You could try laundering the toys anyway if you believe that they would be suited for it despite their label. Of course don't risk it with the most precious toy...
Usually though, nonlaunderable (is that a word?) toys have instructions how to care for them. That most often boils down to wiping the surface with a cloth moist with lukewarm water and mild soap.
While this will (after some repetitions) remove the most obvious dirt from the surface, it won't get rid of the dirt that's stuck deeper in the plush, or in the filling. For this kind of dirt, you're kind of stuck between laundering or doing essentially nothing.
Proper deep-freezing would potentially take care of mites, but it has to be a deep and long freeze - not just the fridge's ice tray. Note that will not remove the mites, just kill them. Those dead mites might in turn become food for new mites later on, and for bacteria. And the bacteria won't be killed by deep-freezing, by the way. Thinking about this gets gross pretty fast.
The best advice to me (although not very useful in your current situation) is to start out with toys that are machine washable, for exactly this reason.