What cards are being added? For Blackjack, if the house adds aces, tens or nines, the house advantage goes down. If the house adds small cards (2s to 7s), the house advantage goes up. Here are the effects of removing specific cards from a 52-card deck on the player's advantage; this is equal to the effect of adding these cards on the house's advantage.
If whole decks are added to form an n-deck game (n = 2,4,6,8 are typical), then the house advantage goes up as decks are added.