I experienced once, and yes it did.
The Pre Burst had been switched into SS automatically to cooling down after a warning on LCD. I was stupid at the time having Pre burst enabled but keep camera idle for quite a long period of time. I was not aware the camera was recording despite I did not fire the shutter button...
It was the only time over heating of my Panny cameras in last 20 years.
As per OP's description, I don't think over heating caused the problem.
Yes I stay away from that mode because of that warning. My default for 4K photo is S/S which I like better anyway because I can set the camera on a tripod and shoot pictures of a perched bird and when it flies away I will get my BIF lol.
Pre Burst and the other 2 4K Photo modes are indeed for different uses.
The 4K Burst or SS, are actually regular video shootings. It is therefore subject to the restriction of usual 4K video recording of the model concerned. It will rarely overheat the camera.
Pre burst is different. It starts recording 4K @30fps when the feature is enabled. The camera will keep the 1" footage (30 frames) in buffer on rolling stock basis and will be saved only we actually shot. After we hit the shutter button, that 30 frames, together with another 30 frames took during the firing in next 1", total 60 frames will be saved to card.
This 1" footage before actual firing is used to compensate for our human response time (see and fire). So we shall not miss the capturing of the happening of an incident. If using the other 2 4K photo modes usually either we have to start the recording well before the incident happen (might get many useless frames) or if we see and fire, we might miss the critical moment of happening.
Pre Burst is very similar to the Pro Capture of higher end Olympus cameras. The benefit of Panny Pre Burst is any 4K supported camera and any AF lens can enjoy it without buffer size restriction.
The issue is once when Pre Burst is enabled, the recording start and will be kept in the buffer on rolling stock basis. i.e. every frame recorded 1" ago will be replaced by the 31 frames later. It means a non stop writing and erasing of data inside the buffer until we actually take the shot. Not difficult to imagine the heat that will be generated by this specific shooting mode. Interesting enough, this will have no built in recording time restriction as 4K video & other 2 4K Photo modes...
30 minutes of uninterrupted photos-- I hope that doesn't heat a camera up?
No matter how fast the Burst (the fastest indeed is shooting with reduced size), the 10~ fps will hardly overheat a camera. Indeed the buffer size and file writing speed will limit the number of usual still burst shots as well. Had you ever succeeded on using Burst H non stop for 30 minutes?
Then I think I know what might have caused the overheating of my camera! The nonstop changing of settings and saving different groups of settings for an hour and then after that taking a few test pictures to see the effects of the new settings. So it looks like the act of constantly saving new information and overwriting old information in the buffer is what causes the camera to overheat?
I did not know that about Pre Burst, I thought it is only activated when you half press the shutter? So it is always on in the background when you activate that mode in the settings?
Yes
At least the problem went away when I turned the camera off and back on-- though like you said it is puzzling that only the shutter button and EVF were affected,
If you shutter release has been set to focused, for some reason when your camera can't set a focus, it will refuse to take a shot. I guess it might be part of your finding.
Pro (no miss shooting) and con (not good for BIF etc) of this feature.
all the other buttons worked fine and the camera didn't turn itself off.
So do you think all is fine and no damage and I don't have to send the camera back?
If not recurrent issue, service center might do no good but just wasting your time.
It wont cause any dead or stuck pixels either?
Sensors of modern days are quit resilient. A few times over heating, if had escaped the over heating protection of the camera, might not affect your sensor permanently. Have you found dead pixel on output?
My first digital camera,Panny LC5 which was 20 years with me, still has a healthy sensor...
No pixel mapping on this camera

I hope electronic shutter doesn't cause dead or stuck pixels either, as without pixel mapping I don't have a way to map them out on this camera, unlike with my Olympus cameras. I'm not sure the electronic shutter on my EM10Mk2 or long burst shooting caused any dead or stuck pixels either, but I do run pixel mapping every few months or once a year.
So far I had done once on G1 after it jumped into a river, well dried and used again a few days later. I did not find any evidence of dead pixel on output, just did it for a peace of mind.
I notice a stuck pixel on the LCD and sometimes multiple stuck pixels, but those are only on the LCD (perhaps the LCD itself overheats if the camera is on for awhile?) But that has no effect on the sensor or other parts of the camera?
Might be, hard to say...
Lol I have not used Burst H yet, I did test the electronic shutter on my EM10Mk2 when I first got the camera I wanted to see how long it took take pictures before slowing down and I found a rate of 7.63 FPS consistently for 15 minutes at full 16 MP resolution and no signs of slowing down using the electronic shutter. At that rate, a 32 GB card would be full in 30 minutes!
Do Olympus cameras also overheat with Pro Capture mode?
Rarely heard because of the restriction on gear used, not always max fps and not many owners actually use it.
G9, has similar feature to Oly's Pro capture, the SH Pre burst, for high speed pure still based shooting. SH can use any AF lens for the max fps. On DPR no overheating was ever noted. I guess Oly might do similarly good since by the end, they are all flagship class models.