#1
Lancake
Normally when you shoot an overheating core, you'll speed up the timer. Unfortunately this doesn't work too well anymore, the timer itself is also inaccurate.

A 1 block ship (a core) that overheats shows you a 177 second timer till it despawns, multiple tested cores despawn at 116 seconds remaining. I can't speed it up so I'm assuming 60 seconds is that minimum remaining.

Tried it out on 500 block ships and I can only speed it up by using beam weapon, even then it acts unpredictable. I was able to bypass the minimum of 60 seconds and bring it down to 0 seconds so it almost immediately despawned and other times I couldn't bring it lower than the visual 177 seconds (so the actual 60 seconds), no matter how hard I tried.
Summary:

Timer shows the wrong number, always despawning at 116-117 seconds remaining.
Only beam weapons can lower/reduce the timer by hitting the core directly
You can often bypass the minimum of 60 seconds with that beam weapon, sometimes you can reduce it to 30 seconds, sometimes even to 0 seconds.

I think this was here for a while now.

#2
Smiling Demon
Overheat timer definitely wrong ..
1- applied enough damage to a core to just overheat it
2- timer starts at 300 sec
3- core despawns always at 240 sec remaining