If a biologist were to find a single tooth from an alien or fantasy beast, what could they figure out from that tooth? Would it be possible to get any idea of what the animal may look like?
It depends on the tooth. For mammals, it is possible to get a pretty good idea of what the organism is like based on the tooth. The shape of the tooth indicates dietary habits, and it is possible to do further isotope studies to get an idea of what it eats and whether it is semi-aquatic (caveat: these types of studies require additional teeth of other species of known habits to perform as a control group). Additionally, because similar mammals have similar teeth, it is possible to identify an isolated tooth to belonging to a specific family or even genus or species of mammal. Most species of fossil mammal are named from isolated teeth.
However, this is only because mammalian teeth are abnormally complex, and we have a good idea of what teeth look like in each group of mammals. With reptiles, which have much simpler teeth, it is usually possible to tell carnivores from herbivores (but not always), but it is not always possible to get a precise identification due to their usually simpler shape. Crocodile and spinosaur teeth have been mixed up, as have terrestrial crocodile, dinosaur, and rauisuchian teeth. Other teeth can be pretty distinctive, modern amphibians have very distinctive teeth and almost everything we know about fossil sharks (including taxonomy) comes from teeth.
Again, this is all dependent on people having some idea of what teeth look like in general. If there is no good analogue at all, people will be completely lost as to what the tooth belongs to without more complete remains. Helicoprion, plagiaulacoid teeth in multituberculates, and the taxonomy of gondwanatheres are all good examples of scientists being stumped due to their being no good living animals to compare these things do.
In your case if it's something like a dragon, people might be able to tell carnivores from herbivores because the physical properties of meat and vegetation select for similar shapes to process these materials, pointy or blade-like structures for carnivores and flat, ridged, or grinding structures for herbivores (except when they don't, as some herbivores retain pointy teeth for fighting and some predatory omnivores will have grinding teeth). If it's an alien with a completely unfamiliar ecosystem and anatomy, people will likely be at a loss.
tl;dr: Yes, if they have some background in the biology of the area and what the creature is can easily be determined based on comparisons with other fauna. If its completely unknown, the best you might be able to get is someone saying "that isn't from anything that is supposed to be around here".