How you as professors would prefer to be informed in this case? That is, how to turn down a Postdoc fellowship offer that you have already signed a contract.
The reason in my case is I get an even better offer at another fellowship foundation. But the fact that I contacted the professor, we applied for the fellowship together, I got it and accepted it, make me feel quite awkward...
I upvoted both of the other answers, but in my heart I agree with xLeitix: academics live and die by their honor in a much more extreme way than is the norm in the contemporary Western world. I should admit that my views on this are more absolute than those of some of my colleagues (with whom I have had occasion to discuss issues of reneging on signed contracts). When I hear people say that it is okay to behave absolutely unethically -- even illegally, but counting on the fact that your former employer is not going to hunt you down and pursue a surely thankless legal action -- because they have to look out for themselves first, it really worries me. Such a person is baldly stating that ethics can be thrown out the window when the consequences are important...which is of course the time when ethical considerations matter most.
What stops a person who has reneged on a job from reneging on a future job? What stops them from engaging in unethical academic behavior in any of a hundred different ways? Beyond the student level, academia is almost entirely self-policing. In my field, if someone releases a preprint, and I respond by saying that I have independently proved the result but hadn't released it publicly yet, there will be some followup questions and discussion, but unless I give them an excellent reason not to, people will believe me. In science, every time you write a paper describing an experiment you did, you don't submit a videotape of yourself doing the experiment, you submit a written description of the experiment...and people believe you unless they have an excellent reason not to. Whenever you submit a paper, it gets sent to a referee, whom you trust not to try to steal the work. And so forth: only a small amount of "justified unethical behavior" brings this crashing down.
Once upon a time I was myself in the position of having accepted -- via an email correspondence, not via filling out a contract -- a postdoctoral position. Just a few days later I got offered another postdoctoral position. The first position was a great one, but it involved moving to Canada. The second position was probably even better: it was at a top 10 math department in the US. (Financially the second position was significantly better.) But the first position was one that I hadn't even applied for originally; rather, my thesis advisor had made some phone calls and the offer came quickly: people had taken some trouble on my behalf. What did I do? I told the people offering me the second job that I was grateful for their offer but had to turn it down because I had already committed to another. In retrospect, I still feel like the second offer might have been even better for me. But a tremendously eminent mathematician went on the line and offered me a job when -- for a few days -- no one else would. Anyone who knows me knows who this person is. I can be a prickly person at times -- I certainly felt like the "loud American" during my 2.5 years in Canada -- and I imagine that this guy and most of his colleagues know me as slightly eccentric but professionally reliable. Having people know you as professionally reliable is really priceless if you plan on staying in the profession: for instance, we later arranged for someone to transfer part of his NSF postdoc from this Canadian department to my own department.
What should you do in academia if you want to back out of an agreement? It's simple: you immediately contact the party you've made the agreement with, you explain why you want to back out of the agreement, and you see what they have to say about it. They're not going to be thrilled, but they can convey to you how much hardship they will actually incur by your backing out of the agreement. If it is only a moderate amount, they will probably give you their blessing and the agreement can be mutually dissolved. If however backing out of the agreement would turn out to really be a significant, tangible, medium or long-term loss for them, then you should honor it. The OP has done something in the comments which he surely does not realize sounds absolutely obnoxious to more senior personnel: he's made all kinds of assumptions that his reneging will not be so bad for his employer. And he's making them in the face of evidence that this employer actually did go to trouble: applying for a fellowship is certainly trouble! If this is an externally funded position then it cannot easily be filled with someone else, and the faculty supervisor may well already have made plans and committed resources for the OP's arrival. Or maybe not, of course. Ask!
Finally, the OP doesn't seem to understand why getting a tenure-track job offer might make a difference. The difference is that every postdoc wants to get a permanent job eventually, and every postdoctoral supervisor and institution wants that out of their postdocs. Therefore, if you ask to renege on a postdoctoral job because you'd like to take a tenure track job instead, it is much more likely that the response will be: "Bummer for us, but congratulations for you!" or "Well, can you come work here even for a very short time? That will inconvenience us minimally, you'll get some further training and experience, and we'll be able to say that we placed our postdoc in a tenure-track job. It's a win-win!" It is however still possible that reneging on a postdoc for a tenure-track job may be unacceptable to the employer. If they tell you that, then you go back to the tenure-track job and say "Look, I really want this job, but I have some prior commitments. I am the type of person who honors my prior commitments even when it conflicts with my short-term best interests. Surely we can work something out?" It is likely that the position can be held open for an extra semester or year.
How to turn down a Postdoc scholarship offer that you already signed the contract for?
You don't.
I am all but a hardliner when it comes to such topics, but signing a contract and then not taking the offer is a major breach of trust. This is not "close" to burning bridges. Unless the other side is the most understanding person in the world, this is going to go down really badly with the other side, and mark you forever as an utterly unreliable person, whose word cannot be trusted even after the paperwork is already filed. Frankly, if it is just about location and salary, I would strongly advise against changing your mind. The only time when this could be acceptable is when you get offered a faculty position instead of a postdoc.
Some Edits:
After some more discussion, reading Pete's answer, and a good night's sleep, here are some additional clarifications to my answer:
Steve Jossup: "I wonder whether they're working on the assumption that in a typical contract the questioner has already contracted to work for the full duration of the scholarship without notice period"
No, that's not my assumption at all. I think it is perfectly ok to quit a two-year scholarship after one year, because something better has come up. However, to me there is a world of a difference between quitting after a year and effectively handing in your notice before your job even started. I do remember that there is a question here somewhere about what the shortest acceptable timeframe for quitting a postdoc is, but I can't find it currently.
Also, the comparison between postdocs and professional contractors is in this context quite helpful. I have been an software engineering contractor for a short time, and I would have never signed all the legalese paperwork for a customer and then immediately backed out because a better-paying customer came around. As a contractor, like a postdoc, you are living off of reputation, and this kind of thing is not good for it.
"Isn't there a chance that the other person would be ok with the OP cancelling the deal?"
Of course there is. Maybe the OP's original postdoc advisor does not care so much about this position anyway. Maybe the original postdoc advisor is really just that happy for the OP that he found a better post. However, in all the discussion so far, the OP has to the best of my knowledge never indicated that he plans on discussing with the other person. This question is all about telling the other person that he has changed his mind. There's an important difference.
Stephan Tarasov: "Why should it "could be acceptable is when you get offered a faculty position"?"
This is really just an amendment of the previous point. Even if you get offered a faculty position, you should not just tell the other person that you're out after all. You explain your situation to them carefully, and maybe they will understand and you will be able to renege without burning any bridges. The probability of the other person being understanding is much higher if the other post is objectively much better than the original one, which would be the case if comparing a tenure-track position and a postdoc position. A better-paying postdoc does not qualify.
Stephan Tarasov: "this should not hurt anyone!"
I am not sure why you would say that. The other person thought she is going to have an additional postdoc for a specified time frame, and now she isn't. How is that not very bad for her?
Stephan Tarasov: "Are you saying you keeping all of your promises all the time?"
I would say I try to. Anyway, a signed contract is certainly more than just a promise.
Anyway, I feel like I have said all there is to say on this matter. Still, I have created a chat room for this question, and should the OP or anybody else want to discuss this issue further, I invite you to discuss it in the chat room. As the commenting escalated a bit last night (surely my fault as well), I will not be answering any comments here directly anymore.
I think the main thing is "as quickly as possible." You can either call or email, but don't wait any significant amount of time. It will be awkward, but too bad.
EDIT: I'll note that when I wrote this answer, I think the question didn't specify that an actual contract had been signed, so I assumed that the OP just meant an informal acceptance. Obviously, once the contract is signed, you're bound to follow its terms, and as Pete and xLeitix say, there are certainly ethical considerations beyond that. I would probably state things in somewhat less absolute terms; while I think Steven Jessop is missing some aspects of academic culture, I think the question of precisely where this obligation ends is a good one. Does signing the contract create a moral obligation to do the full term of the fellowship no matter what the contract says? I don't think that would be most people's position. xLeitix says it's OK to quit after a year but not immediately, (which I think is mostly a convention based on the fact that in the US jobs go on an annual cycle, so a year later an employer can position themselves better). What about after six months? The very fact that there isn't a precise answer shows that it is a judgement call, but you do need to exercise judgement in a way the OP seems not to be. You are, when considering a change like this, obligated to assess what kind of damage doing this switch would cause (for example, to the research of your prospective sponsor) and judge whether the benefit to you really outweighs this. It certainly hasn't yet sounded to me like it would. END EDIT
Rather than just bluntly specifying that you're backing out, you may want to lay out the situation with the first professor, and ask if there is a way to reconcile things (maybe you can spend some time in professor 1's lab with funding from the second fellowship?). I think it will "save face" if you can think of it as some kind of hybrid of the two positions rather than you simply dropping the first professor.
I would also think carefully about just how much better the other fellowship is. I'm not an absolutist about changing your mind after accepting a job, but it is a pretty major step and one only to be taken if the other possibility is much better. You run the risk of seriously burning bridges, especially if you don't have a compelling reason (say, strong personal considerations, or a permanent job vs. a temporary one). You don't indicate what the difference is, but if it's just a bit more money or slightly better location, then you should probably just suck it up and stick with your original commitment.
