use a resettable in line fuse until you find the issue and make sure you check anything and everything related to the low volt system, sounds like a bare 24 v wire that is rubbing on metal or another bare wire next to it that goes common..Ive found many outside in the condenser cabinet especially on heat pumps where the wires have been tied to copper pipes eventually rubbing through.. keep looking and you should find it , id say look at running new aire from stat to AH as its happening in heat and cool or its in the Tstat itself..
Yep sounds like a direct short as already stated. I had the same problem a few months back. I put a Lil' Popper in and blew probably 12 fuses before I found it. It was an older house with terrible wiring. When I finally found the problem, someone had spliced wires about 15 feet from the panel and just taped it up with electrical tape. I assume they didn't have enough wire to finish the run, but that didn't make me any less upset. The house could've very easily burnt to the ground.