How do archaeologists address time dilation when analyzing carbon dating results?
For all intents and purposes, the Earth represents one frame of reference, as @kwinkunks states. Therefore no effect.

Yes the gravitational field on top of a mountain is slightly less than at sea level. This is very small though. Variations in gravity over the Earth's surface typically vary at most by a few hundred milligals (ie. "one part in 10,000" order of magnitude).

I doubt you would be able to measure General Relativistic effects due to such small field changes (even integrated over 10,000 years) - and even if you could, they are much smaller than other sources of error (e.g. the atmospheric C14 curve, and sampling errors).