Very few casinos allow players to touch playing cards, let alone deal them. There are a few exceptions to this rule, however, that are worth mentioning because it is well known that player interaction within a game makes things much more exciting.
Blackjack - You can"t touch the cards, but you can choose whether or not you want to hit, stand, split, surrender, or double down. Many like blackjack for this reason, opposed to slot machines where you are at the mercy of the reels.
Baccarat - A favorite among high rollers and Asian gamblers, Baccarat is a slightly slower paced game. The fun part comes when the shoe (unit holding cards to be dealt) is passed around the table for the start of each hand. Players are allowed to pull cards from the shoe and play them, or give them up to the banker. Also, like Blackjack, you can choose to take extra cards to better your hand.
Pai Gow, Caribbean Stud, 3 Card Poker, and other poker game variants - These games let players hold their own hands. Though in some cases you are not allowed to exchange cards - or draw - to improve your hand, theses games are much more exciting because players can show their hands to others in order to figure out if the dealer has a better group of cards.
The only other popular game in the casino where players can directly interact with gameplay is Craps, where the "shooter" or person whose turn allows them to throw the dice, is in control of establishing the flow of the game.
Player interaction always makes games more thrilling, and especially profitable when you play your game well!