I started climbing like a lot of people do, in the Boy Scouts. But int he Boy Scouts you don't really climb much. Instead you focus on rappelling.
We often would rappel a steep 150-foot face ten or twelve times a day. Most of the time we would go from the top to the base in a matter of seconds, in two or three long Hollywood jumps down the face. At the base, the belay device would be smoking hot; so hot it would be difficult to remove from the rope.
The question then is, can a hot belay device damage a rope?
A couple of canyoneering guides took to the road to find out.
In review, the team pulled a rope through a tube style device with a car going 30 miles per hour and were only able to bring the device's heat up to about 200 degrees Fahrenheit. Two hundred degrees is not hot enough to damage the rope. The guide in the film then did point out that on a hot day in the desert, it might be easier to bring the temperatures up higher.
The team then completed a second experiment to see at what temperature the rope would get damaged. They found the following
Kevlar Canyoneering Rope - 840 degrees Fahrenheit
Nylon Climbing Rope - 460 degrees Fahrenheit
Polyester Semi-Static Rope - 490 degrees Fahrenheit
The likelihood of a device damaging a rope from heat is pretty small. the guide in the video does come up with a scenario where it could happen, but his scenario requires a lot of stars to align. There is no evidence that something like this has ever happened.