Paladins aren't immune, because lycanthropy isn't a disease.
Lycanthropes and lycanthropy are described in the Monster Manual (p. 206):
One of the most ancient and feared of all curses, lycanthropy can transform the most civilized humanoid into a ravening beast. In its natural humanoid form, a creature cursed by lycanthropy appears as its normal self.
Curse of Lycanthropy. A humanoid creature can be afflicted with the curse of lycanthropy after being wounded by a lycanthrope, or if one or both of its parents are lycanthropes. A remove curse spell can rid an afflicted lycanthrope of the curse, but a natural born lycanthrope can be freed of the curse only with a wish.
The description continues in much the same way. As you can see, lycanthropy is repeatedly described as a "curse", not as a "disease". As such, the paladin's Divine Health feature provides no protection against it.