A room fan or any fan can catch fire.
However it's rare that a fan catches fire as long as you don't have the cords overloaded, have a good outlet, have a good cord and your electric room fan is in good condition.
As long as you have a good fan, have a good non frayed cord and a good outlet there should be no worry about a fan catching fire.
Fans are meant to run for several hours and even overnight without any issues.
I have my electric fan running all night without any issues.
When it's hot I turn down the AC and then turn on the fan and the fan not only keeps me cool but helps me sleep.
I always make sure to check the cord and outlet to ensure they are in good shape and never had any issues.
Anything you use in your home that is electric could catch on fire but if they are in good working condition they should not cause a fire.
Even a window AC can be left on for hours and hours without any worry about it melting or catching fire.
A window ac and a fan usually has thermal switches that shut off the motors and ac compressor when it gets too hot and then after it cools it comes back on.
I've been running an electric fan all day and all night for several years without any issues.
I've left my box fan and stand fan on for my cat and dog when away to give them airflow.
Never had any issues or any of them catching fire.