Water regulations in Switzerland are very strict and most tap water is of impeccable quality. There is also a law which states that any fountain, which doesn't have regulated drinking water, has to be declared as such, so if you see a fountain with no sign on a village square, it's very likely safe to be drunk. This is most likely the reason why there is a sign at those huts.
The situation in mountain huts is a bit different. They are very rarely connected to any piping and the water most often comes from a local spring. In some cases, especially in very high altitude locations and in winter, there might not be any source available, and there is places where drinking water has to be brought up (mostly where cable cars are available).
So most probably the water comes directly from a fresh water spring (a big percentage of Swiss water reaches the taps unfiltered from some spring). Probably the quality is not monitored however as it would be with public tap water, so it can get tainted (after a lot of rain), or be from a non clean spring. Note that if it's melting water (from glaciers or snow), then it will generally also be unsafe.
Providing clean water at such remote places often comes with a lot of expenses for the owner. There are certain places where restaurant owners will charge you very high prices if you order tap water at the restaurant.
So, you will probably be safe drinking the water. However, there is a small chance that the water might be tainted and unsafe. The only one who will be able to tell you is probably the owner.
Note, if there are locals around, you can ask them and they might know of alternative source to fill up your water bottles. As a kid, (I grew up in the Swiss Alps), we always knew of a couple of springs near my grandparents chalet where it would be safe to drink from.