Why does my mac mini heat up and shut off, beeps 3 times continuously.

First off the beep is trying to tell you something! See if this helps If your Mac beeps during startup

But I don’t think that’s your only problem! It does sound like the system is overheating as well! I would install this great App as it will allow you access to the thermals internally as well as see what internal sensors are doing TG Pro often times the free eval version is all you need, I think you need the full version to access the diagnostics.

If you have a sense when the system is about to shut down quickly grab a snapshot of the Apps window making sure all of the sensors are seen (may need to reshape the window a few times to get everything in). Post it here so we can see things as well.

This is most likely an issue with the RAM, which can be seen here My Mac mini is beeping 3 times and not tu… - Apple Community.

It sounds like your computer might be overheating and causing an issue with the RAM connections.

Guides such as Mac mini Late 2012 RAM Replacement - iFixit Repair Guide may help. See Mac mini Late 2012 Repair Help: Learn How to Fix It Yourself. for other repair guides if needed.