I have a PC that has been running fine, except with non-functioning front USB ports. Below images explain why. My question is: Can I continue running the PC like this (provided that it's been running like this for weeks) or is there a reason to change the motherboard immediately? Would prefer to do a proper upgrade at a later stage rather, if possible.
Looks like you had a short in a power supply line near that header, which overheated things considerably. Eventually it got so hot that things separated enough that there was no longer a short. But something must have been limiting the current somewhat - otherwise the power supply would have shut down.
My concern would be that there's no reason (given that things there are just not mechanically stable) that ordinary thermal cycling couldn't cause things to move around enough that it wouldn't short out again, and maybe do more damage.
In other words, I'd replace it as soon as possible.
Well, I'd say it is unlikely that it would catch fire. The traces have already burned away as much as they will. And even if they did short-circuit again for some reason, it didn't burn too bad the first time; the PCB material is called "flame retardant 4" for a reason.
I'd say if you are planning an upgrade soonish, you can keep running it in the meanwhile. It's always a good idea to clean any accumulated dust from the enclosure and to check that your fire alarm works, but that applies equally for any PC.
The actual fault possibly resides in the USB connectors in the chassis, so in addition to the motherboard you should change also the enclosure or at least leave the USB cables unconnected.