If the clay (terra cotta) pots are lined, (painted/glazed), there is no real problem other than weight and fragility.

The fine roots worm their way into the tiny spaces in the clay's surface and as Dutch said...it makes it a pain trying to slide them out for transplant.
The clay (terra cotta) can 'breethe' too much, (drying out too quickly) plus it can absorb nutes and salts throwing-off your application rates and soil chemistry. (the clay can act like a sponge)
Also, you have to be much more careful when handling the clay pots, as they are heavier and more fragile than the plastic ones.

But in a pinch, if it's all you've got...they do work. Using a knife around the inside-edge to unfasten the roots before transplant may be necessary, but some might still be attached to the bottom of the pot so care must be taken.