First some general dangers not related to melting:
Burning your hands braking on the rope (can also happen with semiautomatic descenders like a grigri due to reflex)
Uncontrolled impact on the rock
Without a backup knot (e.g. prusik) and with a passive descender (e.g. tuber, eight) you may let go of the rope due to the heat induced pain or when impacting on some unexpected obstacle.
Grigris do not have a panic function, meaning when you pull the lever all the way there is no more resistance. This has lead to incidences.
The problem with heat is mostly discussed in caving and canoing communities, as there are commonly huge distances to descend. Still the best source I found was Black Diamond, whose engineers have done some tests on this. They looked at damage to the sling holding the descender device, but the results are still relevant.
Some introductory information:
For argument's sake, Dyneema, Spectra and Dynex are all the same thing. Basically different brand names for UHMWPE (Ultra High Molecular Weight Polyethylene), which has a melting temperature of around 145 °C (293 °F).
Nylon has a melting temperature of around 245 °C (473 °F)
If something is in the neighborhood of 70 °C (158 °F), it's basically too hot to touch
A typical "hot knife" used at climbing shops to cut accessory cord usually tops out at over 650 °C (1202 °F).
Skin burns at 100 °C (212 °F)
An old assumption in climbing is to spit on your belay device. If it sizzles, the assumption goes, then it's hot enough to melt your rope, slings, etc.
In simulated situation with ATCs and 181kg weight they reached a temperature of 256degC, with 114kg and a thin rope (8mm) 170degC. To see whether this is reproducible in a real situation they did a field test. They measured at the bottom of a 150foot (45m) rappel and 10mm rope. When descending at what they describe "kamikaze speed" the 100kg test person reached temperature of 130degC. To test longer distances they used a fishing rod to get the device up again fast. They did six runs with the following temperatures: 111, 110, 131, 120, 125, 135 degC. I do not agree that this means they really peaked at 135degC (i.e. it would not heat any further), but the device could cool off significantly between runs.
Next they tested at what temperatures a belay device actually damages material. On Nylon (polyamid) marks on the fabric are seen at 300degC and it has been cut at 325degC. As heat radiation increases with the 4th power of the temperature and real testing only reached temperatures of 135degC max, I do not expect that you really reach such temperatures. The real danger is that you burn yourself when handling the device after descending.