Koponz
I have:
 
i7 2600k
16 GB DDR3 memory
HD 6870
Silverstone ST75F-P 750W PSU
 
I think that should be all of the major components.

Your new 1080 needs more power than your old card, but the PSU seems to be capable enough unless you are overclocking the CPU like mad, are using some liquid cooling kit, have way more fans than a normal system, etc ...
If you are overclocking the cpu, I would try to start the system with stock clocks first. If that does not work, second thing I would try to do is test the gpu in another system just to be sure it works.
 
PS: Just noticed your motherboard has lots of power-saving features (EPU, VRM disabling abilities, etc). You might want to try to turn all of that off while you try to get the system working, just in case.
 
PS2: Just noticed this:

Koponz
The Asus logo appears, I get the usual POST succeeded beep, but instead of starting to boot into Windows it just hangs and gives me a short, quiet beep about once a minute, which according to the manual means "VGA detected, Quick boot set to disabled, No keyboard detected" which doesn't really make sense. And I'm sure I didn't break anything because I swapped back to my old GPU and Windows came right back to life (and yes, the keyboard works), then swapped to the GTX 1080 again and got a frozen UEFI and beeping again.

 
If you got the "POST finished successfully beep", then the beeps you are hearing after are not coming from the motherboard POST (because as you said, it was finished successfuly). So that beep you are hearing once a minute is not the "VGA detected, Quick boot set to disabled, No keyboard detected" from the BIOS. It might very well be windows. Did you uninstall all AMD drivers before uninstalling the old card?
 
This is all I can think for now. Good luck!