Mine have been around oak and acorns. In fall, leaves from oak and various trees get raked up and place in the run.
That said, I've never just placed branches in the run, so can't say if they ate a lot of the leaves that you may not see some type of effect like loose stool or something.
I'd say unless they just stand there and eat all of them, it should be fine. Once the "novelty" wears off, they likely will just use that area as a place to hide and hang out.
If you're really concerned, then perhaps place the branches on top, hang them higher or even on the outside of the run which may provide some shade but would limit their access.