Coat the surface of the plastic with WD-40. Concentrate the spray on areas with the screen printed logo. Allow the WD-40 to stand for one to three minutes on the plastic and then wipe it off using a clean cloth. The WD-40 causes the ink to soften and wipe away.
Apply a small amount of Goo Gone stain dissolver to a soft cloth. Allow the Goo Gone to sit on the ink for one to three minutes. Rub the remaining ink off using the moistened cloth. Wipe the Goo Gone with a clean cloth. Repeat this process until the logo has completely disappeared.
Add liquid dish washing soap to a small bucket of water. Dip a sponge in the soapy water and wipe the surface of the plastic to remove all chemical traces of the WD-40 and Goo Gone. Allow the plastic to dry before applying another silkscreen image to the surface.