I believe I have found a problem, the solution to which can be verified in 0 time (the solution can only be located in non-zero time, however). As a result, the ratio of the time required to locate a solution to the problem over the time required to verify its solution is infinite. I was wondering if this ratio generally has significance in computer science.
There certainly is significance -- and there is a lot that is known about the subject.
If you are willing to make reasonable assumptions, such as the security of cryptographic primitive, then it is straightforward to achieve such a ratio: any secure encryption scheme will provide one example. Encrypting takes polynomial time if you know the key, while recovering the key if you don't know it takes super-polynomial time (assuming the encryption scheme is secure), so their ratio tends to 0 as the security parameter goes to infinity.
If you want to avoid making any unproven assumptions, then I believe it is an open problem to find such a problem. In particular, it is an open problem to prove that give an explicit example of a function on $n$ bits such that it takes a circuit of size more than $5n$ to compute the circuit. See e.g., Is there a better than linear lower bound for factoring and discrete log?. This corresponds to a constant (strictly greater than zero) ratio. So, I don't think it is known how to exhibit a problem such that it provably has a ratio that goes to zero in the limit.