You do if you download it from the website, but not if you download from the desktop app.
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itch.io has a desktop application (download from itch.io/app). It recognizes the games you've downloaded from the application and has the ability to one-click update a game if there is a new version available. Unfortunately it won't recognize it if you didn't download from the app, even if you move it to the appropriate folder, so you would have to redownload once from the app. But then updating is easy from then on. (And Ren'Py is quite intelligent with saves, so you'd actually keep all of them even through any re-downloads.)