Grease is slang used both outside as well as inside casino establishments. Most commonly, it is used to refer to greasing someone else’s hand or palm with money to smoothen out some shady activity or deal. In other words, simply put, grease is slang for a bribe.
When it comes to casinos, if someone is mentioning grease, they are most likely referring to bribing the dealer to conceal illegal activities. It should go without saying that any illegal activity, as well as bribing, is frowned upon in casino establishments.
Therefore, if a player is caught bribing the dealer, both the dealer and the player might, and most likely will, be in trouble. In most cases, the dealer will be fired on the spot, while the player may end up being escorted out and even banned from the casino.