I didn't want to insult my hosts here in Taipei by asking if it's safe to drink the water since it's very developed here. On the other hand you can't flush toilet paper, so not everything we think of as "developed" applies here.
My hostel actually has a tap in the kitchen for filling water bottles that I'm pretty sure is coming through a filter. But when I'm out and about I don't want to always buy water when I see a tap (faucet) available.
In any case the hostel where I work in Australia also has a filtered water tap but all tapwater is fine to drink - the filter just improves the taste of the water.
So what's the story with tapwater in Taiwan, especially once I leave Taipei and venture to less busy areas like the east coast?