Hobbyists call this process weathering, although in this case it is more like aging. You can make a new stuffed animal look old for display sets and events.
There are two types of aging. One is temporary and the other is permanent. Choose carefuly what you want to do beforehand.
For a temporary aged look, you can dress the toy up. Find or make some old and scuffed up clothes, ruffle them up a bit more and dress the stuffie. Add a hat or a bandana, too. Ruffle the fur of the toy too.
Depending on the look you are going for, you can add some accessories to the toy as well. Use your imagination. Replace any new accessories like ribbons with older ones, or scuff the new one up a little.
If you want a permanent old look of your stuffed animal, then you will need to be a little more direct in your approach. You can age lighter colored fur if you dab a just used tea bag on it. You can make the paws look more worn like that, as if the toy has spent a lot of time walking.
Coffee is another alternative. Either way do not overdo it. With aging and weathering it’s the little details that make the big difference. Plus the wear and tear is not even. Start very small to test and then use more.
If you want to achieve a more battle-worn look then take a fine sandpaper or emery board and lightly sand thin patches here and there. Do that on the face and around the nose too. Again, don’t over do it.
You can also trim a little fur from some edges like around the ears. Be careful not to cut into the seams.
Use these options if you are going to a themed teddy bear event for example. Or if you want to make an interesting diorama type display that features a stuffie in some sort of a scene.