iPhone XR not charging and overheating

I'd suggest testing your phone with a wireless charger, as that will give you valuable information whether its the charging circuitry on the motherboard or the connection between the charging port and the logic board. If wireless charging works correctly, that would verify that the battery and charging electronics are working and would leave the lightning port flex cable assembly as the probable culprit.

If charging wirelessly gives you similar results to wired charging, then you may be looking at a motherboard issue, which is generally going to be difficult and expensive to repair. It could, however, also point to a possibly faulty replacement battery, or even a damaged FPC battery connector on the logic board.

Out of curiosity, did you do the battery replacement yourself, or did you have that shop (or a different one) do the work? I'm curious because as of the iPhone XS and later, (including the XR), Apple started pairing the battery to the logic board. Unless you're an Apple authorized repair facility, the only way to replace the battery and not get a warning message that your battery may not be genuine is to transfer the battery's BMS board from the old battery to a new cell. If they did that, then it's possible a faulty Battery Management System circuit could have been just moved from the old battery to the new one, bringing the problem with it. If you're getting the Important Battery Message, then that has not been done and you undoubtedly have an entirely new battery - not that it precludes it from being faulty, but it's unlikely that a new battery would have exactly the same problem as your old one did.

So I'd say your next step would be to pick up or borrow an inexpensive wireless charger and see how the phone behaves when you use that. Test it out then come back and let us know what you find and we'll figure out where to go from there, depending on what you find.