It doesn’t burn, because there’s no free oxygen to support combustion.
It doesn’t melt, for the same reason that ice floating on liquid water doesn’t melt: it’s not hot enough. The Earth is constantly losing heat from its surface – as you go deeper below the surface, e.g. in deep mines, it gets noticeably hotter (I have been to the bottom of Boulby Mine, which is 1100 m deep; I can personally report that it’s very hot down there). Below a certain depth, the temperature is high enough for the rock to melt; above that depth, it isn’t.
Rocks of the Earth’s crust *do* melt at subduction zones, where one plate is pushed beneath another (e.g. off the east coast of Japan) – the plate that’s pushed underneath gets hot enough to melt, and does so. Conversely, at mid-ocean ridges, rock from the mantle is pushed upwards, and cools enough to solidify.
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