For what its worth, my 386 has a coin attached with no diode and just gets mildly warm, I would suspect that if nothing else it may affect the lifespan of the battery.
I never did figure out why the battery in my old P60 got hot while the machine was running. It was a modded RTC so it should not have been connecting to power anywhere so far as I am aware, I feel kind of bad for bashing the guy who modded it. I assume the RTC was faulty (possibly from being modded somehow) because it gets unlimited 5V across those 'pins' when the system is on, I ended up taking the coin out because it burned to the touch, the internal battery had gained a charge too and it held out until I retired that system in favor of the one I have now - parts from the same guy, but no RTC and a much neater mod with no charging problems. The strange thing about that though, was that the RTC only became "Live" in the Batman board, this did not happen when I borrowed it for an ECS some time later and randomly stuck the multi-meter across the battery to see what was happening. Once again though, the current P60 is completely devoid of any issues and I'm actually thinking of moving it over to SCSI soon, also have a 66MHz (FDIV) chip on the way for it, oddly that one (I think, might have been a Socket 7 I was playing with around the same time) appears to disconnect the battery when the system is running.
My 486SX used to connect the battery with no diode directly to the old barrel's terminals, being larger AA batteries they could stomach it but still got warm and leaked some time before their expiry date. After replacing the board I used the External Battery header and the board doesn't feed power back up the wire from there.
Actually, looking at your board, there is a jumper to the bottom right of the image which appears to be "Ext Bat" with a glass diode next to it. If it does, you should be able to remove the jumper and connect the battery to Pin 1 and 4 (1 is positive, 4 is neutral. The middle pins, 2 and 3, are left disconnected in this state, the jumper that is usually there is to select the internal battery you would have removed) thus avoiding any potential problems. You may need to use a 4.5V pack if this doesn't work. If you are going to do that, do check the connector is labelled for this first though, it probably isn't wise to start connecting power to random jumpers.