APs do not need a handstamp UNLESS you are planning on entering a 2nd park (or re-entering your 1st park) within an hour of your previous entry.
Park Hoppers, Park Hopper Plusses, and Ultimate Park Hoppers ALL require a handstamp.
Reasoning: If you run your ticket through a second time in the day, our computer will show us "Already Used". We (turnstyle CMs) now know the ticket has been in a park, the hand stamp makes sure the same person is using the ticket all day. APs thus don't need a handstamp because they have the biometric finger scan as ID.
(On a similar note, the FL resident passes (play 4 days, and season pass) that have a biometric finger scan also don't need handstamps)
Always get a handstamp if you are planning on multiple entries within an hour, because the turnstyles will reject the ticket as being "recently used", and not even get to the point of the biometric scan. Seeing a handstamp allows us to override it.
And yes, we do check handstamps. The blacklight is positioned such that we can often see a handstamp without specifically asking for it. And yes, you could be denied admission if you have a pass that needs a handstamp and you don't have the stamp.