Is it possible to prove that, for a given problem, no optimal greedy algorithms exist? Greedy is a non-formal term, but it could be (not sure, that's why I'm asking) that for certain problems, greediness can be mathematically formulated and thus be proven that no optimal greedy algorithm exist. Is this possible?
Yes, there is such work. Allan Borodin with coauthors developed a theory where they formalize the notion of greediness and obtain results which approximation ratios can be reached with them. They introduce a class of priority algorithms, which generalizes greedy algorithms. Their first work on this topic is 
the paper "(Incremental) Priority Algorithms".

P.S. The previous paragraph answers on a different question: Is it possible to prove that, for a given problem, no greedy algorithms exist with approximation ratio less than some $1+\epsilon$? What concerns the question I suppose it is assumed there that optimal means exact so the question relates to problems in P (I assume greedy algorithms have polynomial complexity though I guess this is not necessary) which are known to have solution by other methods than greedy ones. And I don't know the answer on this question.

To ivotron: if you didn't mean my first interpretation I will delete this answer.