I don't like tire shine for this reason, it leaves a mess. Not only that, unless you don't drive the car for a while after, the wheels just sling the greasy stuff all over the sides of the car.
Best way to apply tire wet or shine stuff is to put it on an applicator pad and apply it to the tire. Don't spray it on the tire directly. If you do it in coatings after letting them dry, it tends to wet the tire a little better and make it look pretty nice.
Tire shine, IMO, is good for showroom cars and garage queens, but not very practical for driving.
As for getting that off, SONAX will probably do the trick.
'11 BMW E92 ///M3 - ZCP and DCT
'13 Toyota Tundra - 4x4 Platinum CrewMax, 5.7l iForce V8