Why is my phone getting very hot and turns off every 3 minutes?

Is it powering down, or restarting every three minutes? The distinction is important, as a three minute reboot cycle is indicative of what's referred to as a kernel panic; the operating systems kernel detects an unrecoverable hardware error when it goes to scan its sensors (every three minutes), whereupon it logs the error then reboots the phone in an attempt to recover from the error. Unfortunately, if it's a hardware error then the next time it goes to scan the sensors, it sees another error and goes through the whole log the error and reboot procedure. Sad to say, the computer isn't smart enough to know that if it's done this a dozen times and it still isn't working, another time isn't going to help, so it merrily goes about rebooting to infinity and beyond.

The good news is the logs that get stored contain the reason for the reboot and can be analyzed to isolate the problem. We have a whole Wiki page here on iFixit dedicated to analyzing the reason for a kernel panic, and it gives instructions on both how to access the logs and how to interpret them.

iPhone Kernel Panics - iFixit

So your next step is to read that page and go locate the newest panic log. Either copy and paste or take a screenshot of the first page of the panic log then come on back and edit your question to include that information. Once we have that we can take a look and point you to the most likely repair that will fix the problem you're having with your phone. I'm tagging the author of the kernel panics page, Alisha (@flannelist) so they can take a look at your logs as well.

Okay, with the panic log we can see the lines that give us the answer we're seeking.

SMC PANIC - ASSERTION FAILED:
S.sensor array 0 - 6 is 0x0, 0x800, 0x0, 0x0, 0x0, 0x0, 0x0

Those number in the sensor array are error bits; each bit represents a different error that can be detected and logged. So here we see error 0x800; cross referencing to the Wiki page, we find this entry for iPhone 13 models.

• 0x800 is on the charge port assembly.

So there you have it; the error has been detected in a sensor on the charging port assembly. Although it is theoretically possible for the failure to be on the motherboard itself, in practice that is exceedingly rare, and in almost all cases replacing the charging port assembly will fix the problem. It is likely the source of the overheating you're experiencing as well, although you'll have to replace the assembly first then retest to verify that's fixed it; there is a non-zero possibility that the battery is at fault but you can't diagnose that until you've fixed the defective sensor.

Unfortunately iFixit doesn't have a guide for replacing your charging port, but there's a Spanish site called Nadie Me Llama Gallina that has just what you need. It is, of course, written in Spanish, but used with the Chrome browser and its translation feature, it is quite usable and well written.

Manuals / Apple iPhone 13 Pro / Charging Flex - Nobody Calls Me Chicken

As far as the part itself, you can purchase that right here on the iFixit website if you'd like to support their repair efforts.

iPhone 13 Pro Lightning Connector Assembly

Note that it comes in various colors to match the exterior of your phone, although that's strictly cosmetic so it won't cause any problems if you don't happen to match; you only see it if you actually look inside the charging port. If you're economy minded you can also find that part on most of the major retailers such as Amazon, eBay and AliExpress. Here's one that came up on Amazon that should work.

Amazon.com: for iPhone 13 Pro Charging Port Flex Cable Replacement for iPhone 13 Pro Replacement Charging Port USB Charger 13 Pro Dock Connector Board with Tools : Cell Phones & Accessories

As you work, take lots of pictures along the way, then you can use those to create your own iFixit guide to replacing your charging port. It's easy, fun and will earn you the gratitude of those who come after you with the same problem.