If NEC applies, the furnace is supposed to have its own breaker, though sharing with the A/C is allowed. But there should not be other outlets/lights on the circuit feeding those. NEC 422.12 See: https://diy.stackexchange.com/a/236395/18078
Regardless, you appear to have an intermittent connection. Somewhere, a wire (or possibly something else, like the bus-to-breaker connection) is not making solid contact, so sometimes it connects and sometimes it does not. If part of the circuit remains powered, the bad connection is between there and the part that loses power. This can be a large problem because there are generally sparks every time it connects/disconnects, and that can lead to fires - so you need to track down where the bad connection is and fix it.
You appear to say that parts of the circuit remain powered, so it's probably not the GFCI-to-Breaker-Panel connection - a common cause of problems there is using "alien" breakers that might physically fit, but are not actually compatible with your breaker panel and don't connect well as a result.
The GFCI won't trip if the connection is only to wires that should be connected, rather than from them to ground. An AFCI would very likely be tripping, as that's exactly the sort of failure those are supposed to catch.
However, you appear to be saying that the GFCI breaker has failed and does not trip on test, so for that alone it needs to be replaced - as well as finding the bad connection and fixing it. While you are at it, might as well split the circuit so it no longer violates code.
