Can thawing meat too quickly affect its quality? Can thawing meat more quickly (e.g. warm water, microwave, warm oven) than normal (leaving out at room temperature) affect the taste / texture of the meat? Is there a particular maximum rate of temperature change for various types of meat (beef, pork, and chicken is what I'm most interested in)?
I'd be more concerned about the meat's safety than taste/texture. "Warm" thawing creates the perfect environment for bacteria growth.

That having been said, my experience mainly involves the use of a microwave to thaw meat. When microwaving meat to thaw, try to avoid the meat having corners or protrusions. This adversely affects taste/texture because they usually start to cook before the rest of the meat thaws. 

Freeze ground meat, in a circular shape with a hole in the middle for quickest microwave thawing time. When thawing meat 'parts' place them in a circle, thickest side outward.

So long as no noticeable cooking has started, the taste/texture should be fine. Cook immediately.