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&ndash;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.