They will, of course, only eat the parts they can REACH. So if budget is a concern, you don't technically need to cover the upper parts of the walls, or the ceiling, except anywhere that a roost or etc will allow them access.
However you are likely to find you want to eventually cover the whole thing anyhow, b/c it is really not very well cleanable nor durable-to-dents-and-scrapes.
I would recommend the thinnest plywood you can find, or scrounge someone's old 1970s rec room panelling they're removing, or something like that. It's not something you have to buy fancy new material for; it is often one of the more scroungeable parts of the coop.
Most definately! I pen my birds up in one of our garage stalls over the winter because it's heated so it stays a consistant 50 degrees in there when it's -40 outside. The flock decided to perch on the garage door tracks above the opener and had a feast on the foam. Because of the way the garage is set up it wasn't possible to cover the foam with plywood or anything. I am pretty sure that if one was to fall in a puddle from the flooding we have had lately she/he would float from all that insulation they ate.
Card board insulation was suggested to me. If they insist upon destroying it, I would rather it be a cheap fix. We are saving all of the food boxs. Cereal , cookies and what not. And I will be lining the inside with larger thicker cardboard. and a staple gun. I also considered shredded paper. We get alot of crap mail.
Just cardboard stuffed into the wall cavity is next to no insulation at all, although it IS a good rodent and mold farm.
If you want cheap, try shredded paper or dried leaves or wood shavings that you stuff loosely into plastic bags and put the plastic bags into the wall cavities between the studs. It still isn't *much* insulation but it is better than you'll do by just stuffing cardboard in there. Don't expect cardboard sheets stapled over it as an inside wall surface to last long at all unless you live in the high desert or something.