Which public key cryptosystems are not based on number theory? Are there any public key cryptographic systems whose hardness assumptions don't involve number theoretic problems?
Yes. The McEliece cryptosystem using Binary Goppa codes has withstood cryptanalysis to date. Its hardness is based on decoding. I should note that it has been broken for certain classes of codes.

Another common example is the Merkle-Hellman knapsack cryptosystem. Unfortunately it was broken. It is likely possible to build a secure cryptosystem based on the knapsack problem. I wanted to at least mention this line of work.