You wash your chef’s knife and kitchen towels, right? You should wash your oven mitts and pot holders, too.
Think about it. Although we make an effort to find oven mitts and pot holders that are nimble and dextrous enough for you to avoid putting your thumb into a cookie when you rotate the baking sheet, accidents happen. They might get smeared with tomato sauce and melted cheese when you pull a bubbling pan of lasagna from the oven. Or they might accidentally flop into a pot of soup as you move it off the stovetop.
It’s a matter of aesthetics: Dirty oven mitts and pot holders simply don’t look very good. But that’s not the only reason to clean them. Dirty oven mitts and pot holders are a safety issue, too.