Does needing a tune up make your car vibrate while driving?

You should check the motor and transmission mounts. You could also have a bent crankshaft or a balancer problem. But if you feel it coming off of the rear it could very well be the tranmsission mount or rear engine mount. If the mount has failed you must replace all of them as a set or the older mounts will place all of the stress on the new mount and tear it up really fast.