Do routers run on Linux?

Yes most of the router uses a fork of Linux and comes pre-installed by the hardware company. But in most of the case you can replace the firmware that comes along with the router as pre-installed.

What OS do routers use?

The most two famous router operating systems are Cisco IOS and Juniper JUNOS. Cisco IOS is a monolithic OS which means it runs as a single operation with all processes sharing the same memory space.