What do you think did re seating the jamma harness or keeping the door off fix the problem?
Sounds like the amp chip is overheating, this is usually because it is oscillating. It will still work as an amp but will get hotter and hotter until it shuts down or dies. The heatsink should be warm to the touch, if it is too hot to touch then you have a problem. There are two reasons why amps start oscillating, the most likely is high frequency noise on the 12v power rail, this is usual caused is bad or missing decoupling caps on the 12v rail near the amp chip. Usually there are two caps strung between the 12v rail and ground to mop up this interference, one will probably be the largest cap on the board, physically as well as capacity, usually 1000uF or 2200uF, the smaller one will be a small poly cap, in a few nanofarrads in size. If either is missing then you can get this problem, the large electrolytic is the most likely as it is easily knocked and damaged.
The second option is that the signal feeding the amp is picking up RF interference, with arcade PCBs this is unlikely as the audio section and amp is usually on one board, in your case I cant comment as the amplifier is not in the photo - what I will say is that you have the audio cable tied along side what looks like the mains feed - this is a very bad idea, you will get a huge amount of mains interference on the audio signal which can cause the amp to start oscillating.
The fact it runs ok with the door open suggests is just gets too hot with the door shut and is just about ok with the door open.
This isn't a heatsink problem, or a sign you need to fit a fan, the amp will eventually die if it is running long term like this. Simple way to test is to touch the heatsink after the board has been running for 5 mins, if you almost burn your finger then this is your problem.