Not sure about this question. Have thought about it, but not entirely sure whether the structure HAS to be reversible or not.
No, not necessarily however by design it has an inverse. Even the $F$ function doesn't have an inverse, the Feistel network has its inverse by the design.
Invertibility means that we have a PRP. Invertibility is only necessary when you need the reverse of the Feistel network. The archaic mode of operations like ECB and CBC mode requires the inverse since they are designed with PRPs whereas CFB, OFB, and CTR don't require it since they can work with PRFs.
Although, one can modify the Feisthe network a little to have no inverse it is more common to use hash or PRF construction to produce a cipher in CTR mode like ChaCha20, etc.
Working with PRF is better since provides a wide range of functions and eliminates the need for reverse implementation either in hardware or software that helps to reduce the costs.