Possessives of a title in italics

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.

• Chinatown's means the movie name is "Chinatown's".
• Chinatown's means that you are saying something particular to the movie and the italicsation is for purpose.

It's better to rewrite the possessive of something inanimate (like a movie).

Chinatown's plot... => The plot of Chinatown...

This becomes even more evident when the movie ends in an s.

The Silence of the Lambs's protagonist... => The protagonist of Silence of the Lambs...

While this sidesteps your question, if one must choose how to italicize, italicize the title but not the apostrophe.

Please also see When did it become correct to add an “s” to a singular possessive already ending in “‑s”?.

Found a source! - per CMOS 8.170, the possessive after a title of work is NOT italic. They give this example:

The New Yorker’s cartoons generally have nothing to do with the surrounding text.