I've finished a sinew backing on a short osage bow. I've let it dry for over a month. Can I just put a finish over it or should it have a rawhide backing?

I'm not particularly concerned about the bow being in the rain. This was just more of a project to use a leftover piece of osage than a dedicated hunting bow project.

No need for the rawhide. It will just add extra physical weight to the limbs. You can add your finish right over the sinew with no problems. Be completely sure the moisture is out of the sinew, glue and wood before you seal it. Moisture trapped under the finish can begin to rot the sinew and glue.
 I built a copy of Jay Massey's Medicine Bow a few years back and added rawhide over the sinew like Jay did on his Medicine Bow. It worked out fine. For other sinew backed bows I used snake skins which are lighter in weight and give better moisture protection.

I agree, can't see any benifit for rawhide backing a sinu backed bow,unless it was too fast and you wanted to slow it down?John

Like Pat, I use snake skins to cover all my sinew backed bows.  It's a personal preference, but I don't like the look of either sinew or rawhide backed bows.