I am a mathematics PhD student in the US graduating this year. I recently received a very good 3 year postdoc offer in Europe, which I have to accept or decline in early December. Not being a big risk taker, I am leaning toward accepting the offer.
I have also applied for an NSF postdoc, which I would prefer to the European offer but am obviously not a shoe in to get--and the results of which won't be known until February. If I accept the European offer, I will have to decline the NSF postdoc if awarded.
If I do that, would I be allowed to reapply for the NSF next year?
If you accept another position before the NSF Mathematical Sciences Postdoctoral Research Fellowships are awarded, you should officially withdraw your NSF application. The way the NSF postdocs work is that there is no second round of offers: if you are awarded one and turn it down, then it won't be offered to their next choice of candidate, who might have received it if you had withdrawn your application. Furthermore, I believe withdrawing will preserve your eligibility for next year, while you will not be eligible if you decline an offer (according to the official instructions).
I think Anonymous Mathematician has left out something pretty important here, which is that often these situations can be finessed. Have you spoken to the people at the school in Europe about the possibility of you getting the NSF postdoc? Often you can spend one year at wherever you applied to for your NSF and then move to a different place. It's also probably worth speaking to the project officer for the NSF; a lot of things about the fellowship are really up to him/her, so it's possible you could get approval to transfer your NSF to this new place. There's no guarantee, but it's possible with the timing they'll be understanding. You'll never know until you ask.
I completely agree that you need to withdraw your application from the NSF and any other jobs if you know you won't accept them this year. That's just common decency, and it will save people trouble.