Why do I have residue at the bottom of my tea cup? I've tried bottled water and tap water, and the result is always the same. I've tried many different brand of tea bags and they are all the same.
You could try making tea with loose leaf tea and a strainer (rather than tea bags). I have an inexpensive teapot with an insert. It is big enough to make 2-3 cups. There is still a little of the tiny tea flecks in the bottom, but if you pour carefully they stay in the pot and you can have a no-residue cup of tea.
That sounds like what your experiencing. But also keep in mind that the grade of tea used in tea bags is pretty poor unless your springing for high quality stuff. I've found that if your water is too hot, too much tannin can be released and this can cause a different mouth feel, sometimes described as "meaty" or "chewy" similar to some of the desired characteristics found in wines (where tannin content is large).
I'd say experiment with different loose leaf teas as described above but to also work with different water temperatures and water sources. If your bottled water is coming from a source up the road from you the mineral content may not be that different then what your getting from the tap.
Try filling the cup with some vinegar. Just enough to cover all the parts with residues. Let it sit for a few hours. If the residue marks are not too strong, try a mixture of vinegar and water.
This method is well known among Chinese people (from tradition) and it has chemical explanations too.
Try it :)
EDIT: Sorry my English is not good enough and didn't comprehend "residue" well enough. In that case, I suggest you just leave the last sip and don't drink it. Also, some tea doesn't have this problem. But as far as I know, most tea have this problem.