Disadvantages of the Feistel network
Well, what are we comparing Feistel networks to? The other major design paradigm is SPN: https://en.wikipedia.org/wiki/Substitution-permutation_network. In comparison to SPN, the cons are:


Not a direct implementation of confusion and diffusion. Feistel networks have more structure that has no clear justification. (Some point to the security proof but if you assume a cryptographically secure round function, how wrong can things go? A better question is: "if your round function is weak, what does the Feistel network add?").
Parallelism is about half of an equivalent SPN, which is a disadvantage for hardware implementations.