My understanding is that the dryer has a much bigger impact than the washer, although turning things inside out between the washer and drier is a pain in the ass.

Screen prints will last longer because they don't rub on other stuff as much when they are in the inside, especially in the drier where they also touch the hot drum. Some colors will last a bit longer inside out for the same reason. This was a much bigger deal when clothes bled more, but most newer dyes don't have as much of an issue with current detergents.

I don't worry about it except for the screen prints, just whichever way they ended up when taken off.