Which are you trying to burn-in, the phones or the amp?
FWIW, good solid state gear can be left on 24/7 indefinitely. SS gear needs to be warmed up for optimal sound, which is why leaving it on is often SOP among audiophiles. The only SS piece in my main system is a pure Class A 100w/channel amp that I use to power my subs. It has a standby switch which allows it to stay warm at idle (not fully powered-up), but it is still "on" all the time -- the best of both worlds.
If you are burning in the phones, any source will do. So, maybe use something else that doesn't cause so much concern for you. Heck, even put it on a timer if you do not want to run 24 hours a day. My 2 cents.