I just found
this, which summarizes tests conducted by Brecoflex for their timing belts in robot applications. It includes a chart of "friction" tests on carpet. They quote a coefficient of friction of up to 1.6 on carpet with smooth natural rubber.
Some have had success with using their belts as treads; others (1114/1503/1680 in 2006) haven't. Either way, they're an expensive option and require some dedicated engineering to do right.