Why would a tire be for trainer use *only*? Yesterday I installed a Continental Hometrainer Tire on the rear of my CX bike (which had been purchased mostly for trainer use and which came with knobby tires unsuited for that purpose). To my surprise, the sidewall of the tire is marked "for trainer use only, do not use on road." Beyond potentially getting them dirty, is there any risk to using such tires on the road?

Turbo trainers are notorious for quickly wearing out tires. Trainer tires are therefore made of a hard wearing compound. However, this compound may not provide enough traction for safe cycling.

Traction is immensely important as it is usually the limiting factor to the maximum speed in curves. If one's rear wheel slides away while cornering recovery is usually not possible.

See also the related question: What kind of tire should I purchase for use on a bike trainer?