What problems can arise from not following these guidelines
We loose the security reductions (of RSASSA-PSS and RSAES-OAEP) or security endorsements (for others), since they are made under the hypothesis that the RSA key is used only for the operation mode considered. And we know there is potential for problem from actual attacks that can occur when using the same key for signing (any mode), and encryption or signing by textbook RSA (a decryption or signing oracle for textbook RSA can be turned into a signing oracle for any mode).
Also to consider: a poorly conceived signature padding or implementation has the potential to leak the key itself, thus killing the security of a perfectly good system using the same key. However this is not as much to fear with RSA as it is with Rabin signature, where there are a number of attacks on this theme.
What if one uses RSA-OAEP and RSA-PSS with the same RSA key pair?
There's no known attack, but the existing security reductions no longer apply. I'd be surprised if an attack emerged, and not surprised if a security reduction for the combination could be made (e.g. from the reduction for Full Domain Hash), with a lowered proven quantitative security level perhaps.
In any case, it's good practice to stick to the principle: one usage, one key.