TL:DR - Potions, A History of Magic and Defense Against the Dark Arts - possibly Astronomy, Transfiguration and Herbology
It can from the surface be useful in three separate classes, if we delve deeper it can be possibly be useful in six classes.
Throughout the Potions classes in the series not only are they required to know of certain ingredients from plants, but also certain ingredients from magical beasts. It is obvious that Fantastic Beasts and Where to Find Them could be useful in this situation, there may be homework about gathering ingredients or a student may want to know where to get ingredients for... personal use...
A History of Magic
There are a lot of creatures on the grounds of Hogwarts and beasts also play a prominent roll throughout History in the potterverse, there are a lot of discussions in Fantastic Beasts and Where to Find Them about how beasts have been classified and the history of the classification. What was happening around the time of said classification also features, the history of both the Mermaids and Centaurs is very interesting for this reason.
Defense against the Dark Arts
I think it's incredibly likely (given a proper teacher) that this book would be useful in DADA it not only lists Magical Beasts but ranks how dangerous they are. Fantastic Beasts and Where to Find Them could easily feature in DADA for homework assignments or even to learn about deadly beasts!
The Other Three Classes Not as well connected as the previous three
Centaurs feature in Fantastic Beasts and Where to Find Them and are extremely well connected with the stars (Astronomy) the book could be used as a part of research about other races who observe the stars.
Animals often feature in Transfiguration and as such it may be useful to have a good description of an animal for proper transfiguration purposes. Or to cheat on homework by perfectly describing the grindylow you transfigured an ottoman into
Herbology seems to be the final class where it may feature, just for the often close relationship of plants and animals, and also the shades of grey between what is an animal and what is a plant in the potterverse.
The one class that doesn't fit in with the others
I could not think of a even remotely apt link between Charms and Fantastic Beasts and Where to Find Them.