That depends entirely on the game. The games from itch.io I have played so far have been pretty good about keeping saves after updates. However there is always a chance that a dev changes something fundamental which would make it impossible to carry over old saves.

From my experience many (but not all) devs leave notes about how to manually update/patch a game and what will happen to the save files on a game’s download page. If you’re out of luck in that department you could create a new save (or update an old one) for the game in question. Then search for files changed today on your system to identify and manually backup that game’s save folder.