Are there any security issues when replacing the SHA-256 initialisation values?
Random IVs are fairly clearly ok. NIST secure hash standard defines SHA-512/t with arbitrary t and different IV values that are themselves outputs of SHA2, and Merkle-Damgård hashes basically do the same internally when you have a multi block massage. There should be no reason you could not do the same with SHA-256 or with another PRNG generating the IV values.

Nonrandom IVs are theoretically a different matter. Current (partial) attacks on SHA2 work on the compression function and find collisions for any predetermined IV/chaining value. However, at least in principle some values could be easier to attack. That would not break the whole hash as long as the number of weak IVs is negligible.
(As in the case of the Malicious SHA1 linked in the comment by CodesInChaos, but with round constants instead of IV.)