Why wouldn’t you simply sew a tear on a stuffed animal? Well, you may not know how to sew. Or the tear is small and you don’t want a “scar” on your stuffed animal. Or, the tear may happen when you are out or travelling and you have no access to a sewing kit. And you can’t simply leave the stuffie with an open wound, right?
If you are on the move, then treating a tear in a stuffed animal is something like a wound on a person. You want to close it fast and secure. If you really have no access to any sewing kits, then simply bandage the tear until you get home. Use a piece of cloth, a sock or something like that, depending on how big the tear is. You want to wrap the bandage all around the part that has the tear so that the stuffing stays intact.
For small tears you could try the so-called fabric glue. Apply it along the edge of the tear on the one side. Then hold the two sides together by pushing them together and wait a while until the glue dries. This should fix small tears without leaving too much of a visible scar on the fabric of the stuffie.
Another alternative is to use hot glue. It is the same method as the fabric glue, but it dries faster, so you need to work faster. It also hot glue (duh), so you need to be much more careful.
A third alternative is to glue a patch over the tear if its edge is too ruffled or it is too big to be glued or sewed. This might not work for furry stuffed animals since it will be quite obvious. For them, there is a special guide how to fix their fur right here.