Tea water: heat to 80 °C or boil to 100 °C and let it cool down to 80 °C?

It's actually the opposite, you shouldn't boil water for tea unless you want it boiling. Water has dissolved oxygen in it, the more you have the nicer your tea will taste. This has been covered in this question.

The hotter your water gets, the faster it loses dissolved oxygen, so you'll get better tea (for most people's palates) if you raise your water to 80°C and use it right away. If you boil it and then let it cool you will lose much more O₂.

FYI, 80°C is pretty low for most black teas, I experimented with this some years ago and found that most black teas brewed at 80°C came out pretty awful, green tea seemed to be the exception to this. I found 90–95°C to be more of the sweet spot.

Other than boiling to kill pathogens the one thing I can think of would be to purge chlorine from the water, which boiling does. However, you'd need to boil it for 15 minutes to get rid of all of it, not just raise it to boiling. Also, boiling doesn't get rid of chlorinates, which are used to purify water more often these days. See this question for more details on that.

Boiling helps getting the lime out of hard water. In that case, it's better to brew tea with water which had boiled, since lime interferes with brewing process and gives your tea a chalky after-taste.

Edit: Since I got several sceptical comments regarding the statement above, here's a wiki page on temporary hardness, which is due to dissolved lime mineral and can be removed by boiling.

As for dissolved oxygen, I've found an article which claims it has no effect on tea taste, and cites several sources to back up that claim.

There's not much difference in taste between the two options. I do recommend to get a water cooker that can boil it just to 80C for two reasons:

1. It takes 3 minutes or so to let the water cool down from 100 to 80.
2. When you reheat the water twice to 100C to steep tea, the tea is gonna taste dull. I don't experience when reheating to 80C.

Update: I noticed when I answered the question that I assumed use of bottled water. That's what I use, since tap water isn't great in my country. For tap water, I would recommend to always boil to 100C first.