To be fair, most of the problems revolve around time travel, the fundamentals of which are lost on 99.9% of the audience, and event those who are versed in logic and science can get wrong. But of course there are some nitpickers out there.
1. J v. Boris at Cape Canaveral
Normally time travel takes you instantly from A to B. But in this scene time rewinds like on a DVD player. This is not how time travel usually works.
2. Time Travel Launch Velocity
In real-world science, any sort of speed that can promote time travel is faster than the speed of light. Stolen from Back to the Future is the idea of reaching a certain escape velocity. Unfortunately the fall from the Chrysler Building, in which he only just manages to make the needed speed is much higher than J’s fall from the space rocket, where he achieves it more easily.
3. J is Uniquely Special
Somehow the ramifications of changes in historical time have zero impact on J. If K is dead (nobody at MIB HQ remembers him), then how come J is accepted as a staff member, given that it took K to recruit him? Or, more to the point, how come J remembers K if nobody else can. This is physics, and physics doesn’t discriminate.
These plus three more are over at Mike Ryan’s piece at Huffington Post.