As mentioned in comments, this is going to depend on the type of stone and how quickly you want it done.
Sandstone, soapstone, some pumice and a number of other types of stone are less than 3 on the Mohs scale, meaning you can at least scratch it with a fingernail. Most animals with good strong keratin claws can dig through these with a little effort using nothing other than their claws. It'll still be slow, but with some persistence it's not hard... sorry, had to be said.
For tougher types of stone nails aren't going to be enough. Up to ~5 on the Mohs scale you're working with material that's softer than human tooth enamel and the more dense bone structures of the human body. Animals with stronger teeth aren't terribly uncommon, and there are several types of animals in the world that grow new teeth throughout their lifespan. Properly configured this will get you almost as high as granite, letting you tunnel through marble or dolomite.
From Mohs 5 up you're going to need something better than the norm for hard biological structures on Earth. At this point I'd switch tack and look for a chemical solution (yeah, I know, just ignore the puns). Acids sound like a good option, especially as acid production is a routine function of most animals. You'll need to adjust the particular type of acid to better fit the rock type you're attempting to tunnel through, both for efficacy and to minimize the toxicity of the reaction products. And you'll want a decent airflow.
On the plus side you don't have to actually dig the tunnel with acid, just use it to soften up the work surface until it's weak enough for you to dig out the next layer. It's a slow process, but still faster than trying to scratch or chew your way through granite.
For more esoteric options, you can do a lot with ice in a rock face filled with microfractures. In a sub-zero C environment (< 32F for the Americans in the audience) all you need is an organic antifreeze that either breaks down quickly or with some enzyme action. The antifreeze lets water stay liquid long enough to seep into the cracks, then when it breaks down the water freezes and breaks the rock via expansion. Repeat the process until you have friable rock that can be easily cleared to allow access to the next layer.
If ice isn't really your speed, how about something biological that's a bit more explosive? Bombadier beetles mix hydroquinone and hydrogen peroxide and combine the mixture with an enzyme to produce an extremely energetic reaction. While it doesn't get much above the vaporization point of water, up to 1/5th of the mixture of hydroquinone and peroxide can be vaporized. If you could externalize the reaction then it might produce enough heat and pressure to crack the surrounding rock. With the right combination of reactants you could perhaps end up with an acidic compound... or cut out the middle-man and just use the peroxide to eat the rock for you.
Stepping out of the normal biological mold, there are all sorts of creatures in fantasy and science fiction whose natural processes grow crystalline teeth rather than the less durable enamel-coated structures we use. Imagine a process that lets an animal grow teeth and/or claws out of quartz (7 on the Mohs scale), corundum (9) , or diamond (10). Even if they were fragile to impacts and therefore not much use in a fight, they'd still be good for grinding away at the rocks.
And since I've already gone on far longer than you're probably interested, let me leave you with one final option: bacteria. No, hear me out. I always save the weirdest for last...
Your tunnelers have a symbiotic relationship with a type of bacteria that requires specific minerals to reproduce or as a necessary part of their normal metabolism. As waste product they leave behind a compound that either has nutritional value or acts as a drug to the tunnelers. The bacteria grow through the rock following veins of their mineral, cracking the rocks to something the consistency of chalk as they convert the bulk of this mineral to biomass. The tunnelers come along and slurp up the mess, eating the biomass or getting a good buzz, and incidentally spreading the bacteria to new veins in the uncovered surfaces.