Brian,
You might try this for smoke flavored Venison on the Grille.

I use a Weber charcoal grille(dome). Soak your favorite smoke chips OVERNIGHT. Put small aluminum pan of water in center of grille,place coals around pan and light em up. Pan of water keeps meat moist. You may also add your favorite seasoning to water.

If grilling Venison steak,chops,or burgers throw on your smoke chips after coals are good and hot. Put lid on grille for 2-3 minutes,let the smoke build up and then put on meat. Put lid back on between meat flips. In the end you' ll have a very moist meat,not to mention that good old smoke flavor. This is also great way to smoke a Venison roast.

This will all work on gas grille less the pan of water. The gas burners would melt the light weight aluminum pan. IMO nothing like a Weber charcoal grille.