If done before a hard freeze, mice or voles will dig in
and use the area for a nest with your bush/tree being their winter food stores. After a hard freeze, they'll most likely have already made their winter nest elsewhere.
A hard freeze is when the top of the soil is frozen but it can still be easily worked.

I tried this with four tree roses I purchased years ago when I first began growing roses. The first spring I damaged one by cutting ito it with the shovel when digging it up. Another was completely denuded of bark by mice and voles. The other two were fine but repeatly doing this over three years seemed to cause them to lose vigor and eventually turn into high bud unions with a couple of thin spindly canes. I had much better results bringing them into my unheated garage or shed.

I still think it should be unlawful to sell tree roses in cold winter growing zones. They become expensive annuals or require more work to winter protect than most people will do.
I currently have one Polar Joy tree that is suppose to survive unprotected down to well below zero. I haven't tested it unprotected at those temps in the four years I've had it. It's planted in a large black plastic tree pot and wheeled into my unheated greenhouse each winter with a two wheeled dolly. It's gotten very large and blooms well each year. It's well leafed out and blooming by mid April when I move it outdoors again.