I'd like to know whether left or forgotten is correct in the following:
The librarian always goes all around the library before closing time in case someone has left / forgotten something.
I'd appreciate your help.
Both words are fine.
However, if you use left you would normally add behind to the phrase.
. . . in case someone has forgotten something.
. . . in case someone has left something behind.
As a more general comment, it would be more natural to say:
. . . before closing time in order to (see / check) if someone . . .
Using in case someone gives a slightly odd meaning to the sentence.