I once got sick after dinner in a good hotel restaurant in Taipei.

I use bottled water, including for brushing teeth, and never ice cubes. I also avoid salads that might have been washed/rinsed in tap water.

Hotel maids tend to fill the water cookers with tap water. I use it, but make sure that it really gets hot and stays hot for a while before I use the water for coffee or tea.

I've gone to local tea shops in Taiwan, including in smaller towns away from Taipei, with no problems, but I was with locals who knew them.