Even if that marker ink is designed to be permanent, it doesn't have to be. Words doodled on a container lid or scribble marks on a plastic tumbler used to hold markers are equally removable using basic items you likely already have around the house, such as citrus cleaners, rubbing alcohol or a dry-erase marker. Test the cleaning solutions in an inconspicuous area first to make sure they don't discolor the plastic.
A citrus-based sticker and adhesive remover gets permanent marker ink off all sorts of nonporous surfaces, including plastic. Dab a small amount of the citrus solution over the ink marks using a cotton swab, tracing the lines or marks with the wet swab. Wait a few minutes, and then gently dab the liquid with a paper towel or absorbent white cloth. If dabbing alone doesn't lift all the marks, wipe them, following the direction of the ink rather than rubbing across the lines, which may spread the ink a bit. Repeat the process if all of the ink doesn't come up the first time. Wash the area with a mild dish soap and water afterward to remove the citrus solution.
Rubbing alcohol and a paper towel or white cloth pair up for a permanent-ink absorber that lifts the ink away from the plastic. Pour some rubbing alcohol onto the paper or cloth, and press the alcohol-soaked material over the ink marks. Lift the paper or cloth frequently to see if it has absorbed some of the ink; if so, fold the material over and blot the area again. If some of the ink is difficult to remove, dip a cotton swab into the alcohol and rub the swab over the ink lines, following the direction of the marks with the swab. Blot immediately afterward with paper towels. Avoid rubbing across the ink, as this may make the mark bleed outward.
If some of the marker ink doesn't come off after you try other methods, draw over the ink with a dry-erase marker. Wipe both inks away with a dry-erase eraser. Pour a small amount of rubbing alcohol onto a white cloth or a paper towel to wipe the area down if a slight hint of ink remains. This method works best on smooth, nontextured plastic -- a texture may make it difficult to remove some of the ink in the lower areas of the plastic surface. Apply more rubbing alcohol for additional help removing the ink from textured plastic.
Erase the Ink
A melamine foam eraser, designed for a multitude of household cleaning chores, also removes marker ink from plastics. Wet a corner of the foam eraser and rub the damp portion of it over the ink marks, much like erasing pencil marks on paper. Continue rubbing until the ink is gone, and then clean the area with mildly soapy water to remove any eraser residue. Wash the plastic item thoroughly if you use it for foods or beverages.