You can't really have a computational problem that asks you to decide things about functions themselves, which are infinitary objects. At best you can attempt to decide things about descriptions of functions -- for example, you could let your input be a standard description of a Turing machine that computes the function you have in mind.
Here, however, you immediately run into Rice's theorem which states that the set of descriptions that correspond to a particular set of partial functions is always undecidable, assuming that the description scheme is complete and effective, and that the set of functions is neither empty nor contains all partial functions.
So "the set of Turing machines that compute injective functions" is undecidable.
On the other hand, the set of Turing machines that compute non-injective functions is at least semi-decidable, since you can just dovetail simulations of the the machine with all possible inputs in parallel, and halt if and when you have seen two of them terminate with the same result.
We can conclude that the set of Turing machines that compute injective functions cannot even be semi-decidable.