Why is the Fiat-Shamir heuristic deemed secure in the ROM?
The paper by Goldwasser and Tauman is another paper in the series of the failure of the random oracle model; its novelty is applying this principle to the Fiat-Shamir paradigm and interactive protocols. However, an easier place to start is the seminal paper of Canetti, Goldreich and Halevi, called The Random Oracle Methodology, Revisited. This paper shows that there exist signature and encryption schemes that are secure in the random oracle model, but fail for every concrete instantiation, for any hash function. They prove this by carefully constructing a scheme that will fail for any hash function instantiation. Their proof does not work in the random-oracle model (as with Goldwasser and Tauman) since the description of the hash function does not exist when it's a random oracle (it's certainly not polynomial size). 

It's important to understand what this actually proves. The theorem proven shows that it is impossible to generically replace the random oracle with a hash function. That is, there does not exist any hash function that can be used to replace the random oracle in every construction, in order to get a proof in the standard model. Furthermore, there exist constructions that cannot be proven secure in the standard model at all, for any hash function. Thus, the random oracle model is not "sound" in the sense that it cannot always be a stepping stone to a standard-model proof.

What this does not say is that there are no random-oracle constructions that can be proven secure in the standard model. Since my previous double-negative sentence will surely be confusing, I'll explain. There may exist schemes that are proven secure in the random oracle and can be instantiated with a concrete hash function. These proofs do not state that this is impossible. Thus, it is theoretically possible that a hash function can be found to instantiate RSA-OAEP to be secure in the standard model, and it is theoretically possible that a hash function can be found to instantiate Fiat-Shamir on Schnorr's protocol specifically so that the result is a secure signature scheme in the standard model (I may be wrong, but I don't think that this has been proven to be impossible). To be honest, it is very unlikely that this is the case (at least in my opinion), but this hasn't been ruled out.