Legally - all sorts of problems and complications.
Practically - people do it all the time, but take care not to do anything daft like run cables in front of leading edges or (able to go) anywhere near a prop, or take pictures of anything likely to cause official concern such as military bases or nuclear power stations. It it falls off or damages the aeroplane, you're on your own, so make darned sure it doesn't.
The CofA or permit will probably be invalid, whether that concerns you is your problem, not mine!
G