If I allow arbitrary initial hash values for SHA-256, what would be the weaknesses?
Would it be easier to find collisions, preimages, etc?


First note that if you can find preimages for a compressing function (like a hash) you can also find collisions. This means if we can't find collisions, then we also can't find preimages and thelike.

Now, the scenario you are describing is called a "freestart collision attack". Freestart collisions are one of the first signs that a hash functions has a serious flaw, e.g. a freestart collision was found for SHA-1 a couple of years before the full collision. Such a collision would utterly destroy our current faith in SHA-256 and is thus unlikely to be found.



Also note that the quoted passage talks about a variant of SHA-256 that removes the length padding to describe an attack. This attack is prevented by the length padding (and is the main reason why it's there).