I would say there are two meanings: The first is simple. It is just shortening of don't wear it outside. The second means don't use it so much that breaks. We use wear out when things just get old and tired. For example, I had some awesome trainers but they wore out.

Don't wear it out= (out) meaning outside, outdoors. Maybe it's an ugly shirt and I don't want you to wear it out to dinner. Don't wear it out= ( wear it out) meaning to change from a new condition.... Become used or old. I have a new toy and you keep playing with it. Don't wear it out!! Or, I've taught you a new word and you just use and use and use this word. Incessantly. I tell you, don't wear it out!! ( Keep it new) That's all I can think of :) maybe someone else might have other examples.