How you specifically do it will be based upon which software you use. Most photo-editing programs have what is commonly referred to as a clone-stamp tool. It allows you to copy a part of the image and past it over the image somewhere else. It's how you get rid of sensor dust, light poles, cars in the background, old boyfriends and even toothbrush mustaches.
You don't necessarily need Photoshop, but you will need a decent photo-editing app like Gimp (free at gimp.org), DarkTable (free at darktable.org), Lumere (not free) and the like. Use the clone tool to copy nearby portions of the photo over the mustache. As with almost every program, there are plenty of YouTube tutorial videos.