How do archaeologists address time dilation when analyzing carbon dating results? It is a proven fact that the gravitational force exerted upon an object directly affects that objects experience of time; the greater the gravity, the slower time passes, and visa versa. While the rate of radioactive decay may be nearly absolute, the gravitational environment in which that decay took place dictates how time was experienced by the decaying atom.
    Gravitational environment differs with respect to altitude, and would also differ based on the weight and density of material in which an object was located. Furthermore, gentle gravitational waves ripple across the face of this planet every day, creating the weather of our atmosphere. It is also highly probable that the earth has experienced a myriad of intense gravitational events over its history, during unstable periods of solar radiation, and from the ripples of other catastrophic cosmic events.
    I am very curious to read any research in this area.
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).