Removing a patch from fabric is a relatively simple process, which you can do if you pay attention to what you're doing and go slowly. Make sure you don't accidentally tear the original fabric that the patch is affixed to.
Things You'll Need
Locate the stitching that affixes the patch to the fabric.
Insert the seam ripper under one of these stiches and pull up. The sharp edge of the seam ripper will cut the thread when you do this. Make sure to insert the edge only under the stitch and not under the original fabric at all.
Insert the seam ripper under the next stitches all the way around the patch. At certain points you may be able to pull out the thread without ripping each stitch; just don't pull too hard or you can widen the holes in the original fabric.
Cut the knot that is anchoring the thread to the original cloth with the scissors and remove the patch.
Remove all stray threads and lint that have accumulated on the fabric from the patch and thread. You can reuse the patch and sew it to another piece of fabric or discard it.
You can attempt this with just scissors if you do not have a seam ripper by following the same steps above and cutting the thread rather than ripping it with the seam ripper. However, you will need to do this very carefully as it's really easy to tear the original fabric this way, which is why a seam ripper is highly recommended.