Take a can of WD-40 and coat the surface of the plastic item. Concentrate mainly on the area that has the printed logo.
Let the WD-40 sit on the item for a moment, then wipe it off with a clean cloth. This should take off the bulk of the silk screening ink.
Take a bottle of Goo Gone and spread it over the remaining logo ink. Let the Goo Gone sit on the plastic for a few moments.
Wipe the Goo Gone away with a clean cloth. Repeat until the logo has completely disappeared.
Add dishwashing soap to a wet sponge and wipe down the surface of the plastic item to remove all of the chemicals. Let the item dry.
Take the fabric item you have silk screened and run it through a cold wash in the washing machine. Make sure that the wash is cold, as hot water will cause the ink to set.
Lay the item out on a flat surface. Using your hands, pick and peel off any loose paint bits and throw them away.
Put the item into the dryer and run it on the highest setting. This will turn the remaining paint back into liquid.
Peel away as much of the paint as you can. If there are still logo pieces left on the fabric, soak a cotton ball in nail polish remover and daub it onto the fabric.
Wash and dry the fabric again.