"Les Canalettes is a shelter in the Causse du Larzac region of France. It contains Mousterian remains dating to the last glacial, about 73.5 ka. Most interesting is the occurrence of what appears to be burned lignite, which was likely used as fuel and was available as close as 5–15 km from the site, well within the acquisition zone of raw materials. Analysis of wood remains in the site suggests that people used coal when wood was in short supply. Furthermore, experiments by Théry-Parisot demonstrated that the occupants were familiar with some of the burning characteristics of the fuel. For example, adding lignite to a fire which no longer exhibited flames added significantly to the burning duration, thus permitting rekindling of the fire at a much later stage and prolonging the ability to heat. In addition, the study found that a hearth mixed with dried wood and lignite, consumed 4 times less wood than a hearth simply using rotted wood. These results provide important insights, suggesting that Neanderthals exhibited a clear knowledge of the combustible properties for diverse fuels."
So, yes, there is evidence from a one site that Neanderthals used coal, and this evidence suggests they were fully aware of its combustible properties. However, so far it's only been found at one site, which suggests it wasn't a widespread behavior. So, it's not a ton of evidence, but it's certainly suggestive. What's especially interesting, however, is the fact that at Les Canalettes, Neanderthals were well aware of coal's properties. This indicates that, no matter where they lived, Neanderthals could develop a very thorough knowledge of the properties of the various resources that were available to them. Considering especially that Les Canalettes falls towards the later end of the Neanderthal timeline, it also fits in comfortably with the recent conclusions that Neanderthals by that time were regular fire-users, if not paleopyrotechnologists (say that ten times fast!), as based on the review of the evidence recently proposed by Roebroeks and Villa (2011).
Goldberg, P., & Sherwood, S. (2006). Deciphering human prehistory through the geoarcheological study of cave sediments Evolutionary Anthropology, 15 (1), 20-36 DOI: 10.1002/evan.20094
Roebroeks W, & Villa P (2011). On the earliest evidence for habitual use of fire in Europe. Proceedings of the National Academy of Sciences of the United States of America, 108 (13), 5209-14 PMID: 21402905
Théry, I., J. Gril, J.L. Vernet, L. Meignen, and J. Maury. (1996). Coal used for Fuel at Two Prehistoric Sites in Southern France: Les Canalettes (Mousterian) and Les Usclades (Mesolithic) Journal of Archaeological Science, 23 (4), 509-512 DOI: 10.1006/jasc.1996.0048
Théry-Parisot, I., and L. Meignen. 2000. Economie des combustibles dans l’abri moustérien des Canalettes, de l’expérimentation a` la simulation des besoins énergétques. Gallia Préhistoire 32:45–55.