According to my experience it’s no problem.
Some attractions in Prater have their own terms of use, I’ve seen a „giant swing“ requiring closed footwear.
Some museums require shoes in their terms of use, that includes all the „Belvedere“ museums and the museum of natural history. No problem in MUMOK (museum of modern art). In MAK (museum of applied arts) I’ve been warned that the parquet can splinter … which I haven’t observed.