I'm applying for postdoctoral positions and when I get rejection letters there is no reason given. Is there any way I can tell if there are "red flags" in my application that are getting it shot down? My adviser has reassured me that there is nothing wrong with my application but after so many rejections I am not so sure...If there really is a simple reason, I would like to know so that I can remedy my application.
You haven't said how many applications you've submitted or whether you've had any interviews. It's possible that you're just being unrealistic about how many applications it will take to obtain a position.
In the US, employers are reluctant to say anything about why an applicant was rejected because of the possibility that this information will be used against the employer in a lawsuit. It's likely that the faculty involved in the hiring decision will be under instructions to not say anything about why applicants weren't chosen.
Furthermore, even if you could get one of them to talk off the record, the likely reason would be something like "other applicants were slightly stronger or were better fits for our needs." The reason for this is that the academic job market in most fields is extremely competitive for job seekers, and thus most positions attract many (dozens or even hundreds) of very well qualified applicants. You should not expect that your application will have better than a 1 in 20 chance of being selected for an interview.
If you're making to the interview stage but haven't gotten any offers, then it's possible that your performance in the in-person interviews has hurt your chances of getting the job. However, you shouldn't conclude that until you've had several interviews with no offers.
You do not have the right frame of mind. You are asking a question that doesn't have an answer. People who have positions to give don't need a reason to not give it to you. They need a reason to give it to you. When you ask, "why didn't I get the position?", most of the time, there is nothing meaningful to answer beyond "we gave it to someone else". These days, competition is so fierce that for any position, there is a dozen candidates that the university would be lucky to hire. A choice needs to be made. And since we are not robots, this isn't an entirely rational choice. Hiring someone is a very complex decisions with many variables; who is the "strongest" candidate depends on many different things, like the person's skills, whether the hiring party thinks that you will continue to do good research/teaching, whether your future colleagues think you will be pleasant to work with, etc. In the end, a hair can separate the top candidate from the next one, but the choice is binary. C'est la vie.