I know that this topic has been discussed several times, but I wanted to add some information along with my opinion on the topic, based and all what I have heard and read.
Apparently most of the people is afraid from drinking untreated tap water at home. However, normally most of the locals just boil the water and that's already fine for them. Many other people use the very popular RO filters at home, for cleaning the water from impurities and so. However, AFAIK, the water isn't that bad at all at least in Taipei. A coworker, who is chemical engineers, actually states that the water is very good, and not hard at all. The main problem people seem to worry about here are viruses/bacteria AND the high levels of Cl in the drinking water, used for controlling the first problem.
I have tested the Total Disolved Solids of the water at home (Hsinchuang) and at my office, with these results:
1) Office - unfiltered tap water: 40 to 50 ppm
2) Office - filtered tap water: 30 to 40 ppm (the filter components were changed a few months ago)
3) Home - unfiltered tap water: 50 to 60 ppm
These numbers are actually quite good. If I recall correctly, the OMS says that the limit for regular tap water should be something like 400 ppm.
BUT, last weekend I tested again the levels at home, and I got a quite different read: 150 ppm. Not sure if it's due to the rains or what. Today the water at the office had the same TDS value.
BTW, what I would concern the most is heavy metals coming from old pipes and so. Not sure how old a building need to be for having these issues though...