I keep one of my computers on full time and don't worry about a fire.......I worry more about 'wall warts' causing a fire more than I do an internal power supply in a PC.
When running 24/7 though, your CPU will suck in a lot of dust, so write yourself some reminders on a calendar to dust it.
In addition, check your BIOS settings to see if you can get the computer to shut down if the processor temperature exceeds a certain level, or if the fans fail.
Listen to the fans for a change in sound, as this might be an early indication of fan failure.
An odd odor, similar to an open Magic Marker is not a good sign either......Usually indicates a fan has failed and the processor is probably already cooked.
I've been through several desktops in my life......I'd say the ones that failed were either due to fan failures, or power supplies going bad and taking down the motherboard along the way.