The Arduino itself, powered from USB or a wall-wart power supply is reasonably safe.* Heat comes from power (Power in Watts = Volts x Amps). For example, a hair dryer or toaster is often around 1200 Watts. Low voltage is much safer. The Arduino consumes a few Watts.
However, you can get high-temperature with concentrated heat... There are 25W soldering irons that can melt solder.
If nothing's getting too hot, you are probably OK. ...But something might be cool today, and something can go wrong and overheat tomorrow.
If you are switching AC power or building your own power supply, you have to be more careful. A fuse on the power line can help in case of a short or other failure that causes excess current. A fuse won't necessarily prevent a fire, but it will help.
- Of course, you should buy your wall wart from a reliable supplier/manufacturer with all of the regulatory approvals... If you buy a cheap one on eBay or Alibaba, you're on your own...