Do you need to wash your hands to just take out your contact lenses if they’re daily disposables thus you throw them away?

The point is that you're putting your fingers in your EYE. It has nothing to do with the lens. Your eye has mucus membranes, which are breeding grounds for germs and viruses. That's why people keep saying not to touch your face to protect against COVID. Touching your nose, mouth or eyes, if there's a speck of germ or virus on your hand is how you get sick. WASH YOUR HANDS! or use a disinfectant spray if are aren't where you can wash them. 

You should as you are putting them near your eyes