As someone who is not flexible myself, but has flexible friends, I can say that it is very important, not just for finishing routes, but also for preventing injuries.
There have been quite a few routes that I have not been able to do, only to see my friends do them using some sort of crazy body folding beta. Lifting their legs up to the level of their ears, and other moves that I had no chance in doing.
On top of that, certain routes require a certain level of flexibility, and not having that flexibility, I felt close to injuring myself quite a few times while trying the moves.
In general, though, I think flexibility can help even on routes that don't require you to be so flexible. It can allow you to do them more statically and be in control better than if you were not flexible.
I have started training and working on flexibility, and I do see quite an improvement while climbing, though I am still a long journey away from declaring myself as "flexible".