I know they can eat the berries.  Not sure about the leaves though.  I believe when you give them twigs  yo have to dry them first.  They can have hibiscus leaves.  I am not sure about other leaves.

I would be very careful about feeding anything from the yard. Most yards are maintained with some amount of pesticides and unless it’s your own yard and you’re really careful about not using anything like that, even in the soil, it may be risky. Airborne leaves, pesticides from other yards, all may contain unknown elements that are harmful to a house bunny. It depends a lot on where you live. There are wonderful assortments of greens at the grocery and health food stores now. You can give your bunny fresh, clean veggies and fruits that will be very satisfying.

You can feed them things from the yard but you really have to know it’s safe.  I let mine out to graze, they can eat the grass, dandelions, thistle.  I sometimes give rose petals and leaves, violets and weeping willow.   Mulberry is considered safe, with anything though, less is best.  This is a list of non-toxic plants I sometimes refer too.  You can check the poisonous list from this link too.

http://www.3bunnies.org/non_toxic_plants.htm

Note: I’m uncertain if this is non-toxic in general or bunny specific.  I tend to think that it’s ok if a rabbit has eaten a bit of these plants but you wouldn’t necessarily harvest and feed hoardes of it to them.