Why is the title Forget Something? and not Forgot Something?
I believe that it may come from Did you forget something? but this (the title as in the picture) does not sound right on its own. I would rather use the past tense, like in Gone With The Wind.
Yes, it's short for "Did you forget something?". The first two words are often elided in informal speech, so "Forget something?" is a common way to say it. This works for other verbs too: "[Did you] miss me?" "[Did you] finish yet?" And sometimes even for non-verbs: "[Are you] ready yet?"
"Forgot something?" also works, but it's not as informal. That's probably why the person making the sign chose "forget": they want the title to sound informal, because the rest of the message is formal.
As the answer by Anonymous alludes to, "Forget something?" is a bit friendlier and colloquial, which is the overall tone they want to achieve when they are offering to help you and trying to build rapport.
In addition, "Forgot something?" has a bit of an accusatory tone and seems more akin to saying, "[Haven't you] forgot[ten] something?" which definitely doesn't mesh well with the conversational tone the hotel staff wants to take.
