Could installing a higher C rating battery damage the ESC?

[xviper]

If your flight lasts 5 minutes, you average discharge is 12C; using a battery that can honestly provide more than that is not a waste though, as you will get a better burst of power when full throttle is needed, both at takeoff, and at landing. As to EDF use, most of these models that I've seen or flown have less air flowing around the battery than other types, so the higher C rating (lower IR) means the battery doesn't get as hot, especially since many EDF models fly about 4 minutes or less (>=15C). One of the reasons the Graphene batteries stay cooler is that there is a heat-conductive plate between layers, so what little air flow you have gets more heat out of the pack; great for EDF models.

Excellent explanation of what I've been seeing of late. For a couple of seasons now, I've been running Admiral Pro 3600mah, 4s, 50C packs in my SebArt Mig29. I always launch it vertically on full throttle and fly around at ~2/3 throttle most of the flight, except for a couple hovering attempts during the flight. Timer set at 3 minutes. The Pros come down quite hot and feeling a bit spongy but not puffed. The last couple of outings, I've started to use my Graphene 3000mah, 4s, 65C. Same sort of flying, same time but the launch is with much more authority and top speed appears to be somewhat faster. Even the fans sounds a bit faster. The packs come down only slightly warm (if you can even call it that). I've relegated the Pros to my prop planes now, where they perform adequately well. If there was proof in the pudding, the Mig is it. (Same goes for my Yak 130, 70mm - 3000mah, 6s, 65C.)
Just for fun, here's a short clip of the Mig 29 before the Graphene (my buddy just had to do a little artsy fartsy bit with the slow mo on his new Iphone at the time):

[P51mustang]

I have a F16 70mm edf with a 80amp esc can I run a 6cell Lipo battery which has a 60c with a120 c burst

[xviper]

I have a F16 70mm edf with a 80amp esc can I run a 6cell Lipo battery which has a 60c with a120 c burst

If this is a stock power system, then yes. If it was the 4cell plane that you put an 80A ESC, then no. The motor likely won't take it.