I downloaded the demo, played through most of it, then sprang for the newest update. My progress carried over just fine. Like all Ren'Py games, the saves are stored in a separate folder from the game files themselves, so deleting the game folder and unzipping the newer version in its place is the way to go. (Simply unzipping them on top of the extant installation is a good way to bug out a game - don't do this.)
As with any game development, the answer to "will my saves carry over?" kind of ends up being "it depends," simply because backend changes or major revisions can make using old save data difficult and artifacts from earlier versions can get baked into an old save. However, at the moment, saves seem to be working fine.