Possible 3x weaknesses in an Eevee team:
BUG: Espeon, Umbreon, Leafeon.
FIGHTING: Eevee, Umbreon, Glaceon.
The obvious conclusion is don't use either of those trios. I'm not sure if I'm comfortable saying Don't Bring Umbreon; its immunity to Psychic could easily be game-winning. But if you do use Umbreon, don't also use both Espeon and Leafeon, and don't also use both Eevee and Glaceon.
Luckily, there are no other 3x weaknesses possible with an all-Eevee team. Possible 2x weaknesses are Fire (Leafeon and Glaceon), Ground (Jolteon and Flareon), and Rock (Flareon and Glaceon.)