Unless you make the binding pretty wide, I would probably cut off the old binding. If it is going to noticeably clip off star or other points, I would probably attempt to take the binding off instead of cutting... but I can tolerate a few issues on a rescue project. Otherwise you are likely to be pretty thick and lumpy around the edges. Plus, cutting would allow you to fix the batting pile up.
When I was first learning to machine quilt, I took more than one tied quilt from the thrift store and worked with it in various manners, when and how I took out the ties, did projects on both a domestic and a long arm. If it had a nice finished edge, I usually left that intact and left the ties until I was done quilting if I could, or otherwise take out sections at a time in the area I was quilting.