If one writes a word in italics, say the name of a movie, and wants to put apostrophe s at the end to form the possessive, is the apostrophe s italicised with the title? Chinatown's or Chinatown's?
No, you should not italicise the 's.
The reason behind that is that italicisation is part of giving importance or a style to a title, now if you italicize the 's it will become part of the name itself. But if you don't continue the italicisation it will mean that you have added it on purpose.
It's better to rewrite the possessive of something inanimate (like a movie).
This becomes even more evident when the movie ends in an s.
While this sidesteps your question, if one must choose how to italicize, italicize the title but not the apostrophe.