I was doing some cryptography exercises, one of which was an example of a feistel network. It said that the function $f_k$ was a permutation for all k but didn't explain why that makes a difference. So what is the importance of a function being a permutation in a feistel network?
I assume that you are referring to the round function. This is false. The whole advantage of a Feistel network is that $f_k$ does not need to be a permutation. Indeed in DES, it is not a permutation. In an substitution-permutation network it needs to be a permutation so that it can be inverted, but in Feistel inversion is possible in any case. Thus, you have more freedom in choosing $f_k$.