Actually, no, anonymous! In fact, unlike albinism, melanism isn’t tied to any negative health conditions. Rather, it’s actually a favored trait among many pokémon, as in some cases, particularly among species with naturally brown pelts (such as eevee), the added pigmentation allows them to blend in better with their natural habitat, enabling them to become better hunters, foragers, or, well, hiders from predators. Additionally, melanin protects the skin from ultraviolet light, and some studies suggest ties between higher concentrations of melanin in pokémon skin and disease resistance. (Take Alolan rattata, for example. While it’s true that the rattata population adapted to urban life, some studies suggest that part of the reason why rattata in the Alola region are able to coexist with Alolan grimer is because they adapted higher levels of melanin in their pelts shortly before Alolan grimer finally adapted lower levels of bacterial growth themselves. In other words, the rattata that survived long enough to adapt to urban life in Alola were the ones that grew black pelts, rather than the ones that grew purple ones.)
Getting back to your eevee, though, there’s very little difference in the care you’ll need to give your melanistic eevee compared to the rest of its litter. If anything, you may need to keep an eye out for any aggression from the other eevee or other pokémon in general towards your melanistic eevee (as occasionally, members of the same litter that display different coloration—especially shininess—may experience bullying), but this is really only a distant possibility in terms of what might occur while raising your pokémon.
In other words, chances are you should be fine. Best of luck, anonymous!