Can you overmix a gluten-free cake?
I run a gluten free bakery and yes, over mixing is a concern for many cake batters, cookie doughs, pie crusts, etc. I find, what makes overmixing an issue is the gums used in the recipes. This being either xanthan or guar. An over-mixed cake batter will become very stringy and goopy, and will not pour smoothly when run off of a spoon, for example. Cookie dough, when over-mixed, will become very tough and elastic-like. Cookie dough should break apart easily when you separate it, but an overmixed cookie dough will stretch when pulled apart... Kind of like play dough or fondant.

Just like when baking with wheat flour, GF recipes that use gums (so pretty much all of them) yield a tougher, somewhat unpleasant texture.