Does a path exist going through each color only once?
I understand from your question that you are asking for a polynomial-time algorithm for solving the problem. It is unlikely there is one for the problem.

Taking the hint of @jjohn, this is exactly the same problem as in the Kleinberg-Tardos book, Exercise 8.12. They call the problem evasive path. Here, color classes represent different zones, and you must get from $s$ to $t$ visiting each zone at most once.

There are at least two straightforward reductions. The other one is from directed Hamiltonian path, and the other one from set partitioning, i.e. exact cover. If you still care about solving the problem, you can do it in $O^*(2^k)$-time, where $k$ is the number of colors.