Hello,
Good question. I get asked this all the time. This is one of those things that you will just know from experience and time after handling the two felts. The best way is by the feel of it under the fingers and hands. After time you can often tell just by looking and not even have to touch the felt.
Fur felt is by far superior to wool, and has a very soft almost buttery velvety feel to the surface. It is much more supple and pliable and can be easily molded/formed into any shape you wish, with or without steaming. Steam of course will make it even more pliable. It can be reformed, re-blocked, and re-shaped over and over again in many cases.
Wool felt is more rough feeling by comparison and is not as pliable/flexible. It is much harder to block by hand. In many finished hats, it is often treated with some sort of millinery sizing (stiffener) to help it keep the shape after blocking. Once this sizing is added the wool felt cannot be re-blocked or re-shaped with much success. There are some wool felts that were made to imitate fur felts, and sold wholesale, and these (usually made in the 40s, 50s and early 60s) have trade names to make them sound more "soft". Doeskin, Sandswept, Beau-soi, etc. These can be a little more difficult to determine as they feel softer and more pliable, but after time you just instantly "know" by either the name stamped inside or again, the feel of it.
A burn test is usually not needed, and often inconclusive as both are made of animal hair and have similar smells, although there is stronger burnt hair aroma to fur felts. After time you will tell the difference in the smells.