I don't believe the heating effect is significant. After all vinyl is a plastic which is permanently deformed when heated and pressed as a record. At room temperatures it deforms elastically, springing back to shape eventually.
Let's estimate the friction watts converted into heat when a 1-gram stylus tracks a record groove at about 50 cm/second (@antinn). For simplicity let's take a coefficient of friction of 1, so the drag on the platter is also 1-gram. In reality, friction is a bit less, but there is extra drag with modulated passages.
From Wikipedia
At average gravity on Earth a kilogram mass exerts a force of about 9.81 newtons
Let's round a kilogram to about 10 newtons. So a 1-gram mass exerts a force of about 0.00001 newtons.
When an object's velocity is held constant at one meter per second against a constant opposing force of one newton, the rate at which work is done is one watt
The record periphery is spinning at 0.5 meters per second, so the work done by a 1 newton stylus would be about 0.5 watts. Downsizing to one gram, gives 0.5 x 0.00001 = 0.000005 watts, or 0.005 milliwatts or 5 microwatts.

