We use 'mustn't' for prohibitions (things you are not allowed to do) and 'must' for strong recommendations or obligations.
1. You mustn't be quiet in the library. (This is incorrect. Libraries require quiet, so it should be 'must'). Let's correct it: You must be quiet in the library.
2. You mustn't speak in the museum. (Museums usually require quiet to appreciate the exhibits).
3. You mustn't eat and drink at the art gallery. (Eating and drinking can damage artworks and is generally not allowed in galleries).
4. You mustn't run at school. (Running is often not allowed in school buildings for safety reasons).