A vining weed climbed all through a couple tall roses bushes of mine before I was aware of it. It will be nearly impossible to untangle the weed from the rose canes and if I try, it will break off many stems. Is it OK to just trace the vines to their roots and pull out the roots, while leaving the vines to die on the rose canes? Will that cause any harm to the roses?

You are correct in cutting the vine at the root system. Just leave the vine on your rose bush until it has dried. Then through the normal pruning of your roses, most of the dead vine will come out. The speed at which the vine grew hints at it being an annual, which would have died out over the winter.