Actually, until the ball hits the cup or is otherwise unplayable (at the bottom of a water hazard) the golfer has to keep playing the same ball. If it goes into the crowd, they have to play it where it lies, they can’t just give it to the crowd.
At the completion of a hole, however…
Well hockey players have been know to accidently hit the puck into the stands, especially when they are down a man or two. But there was a time when they tried to retrieve the pucks.
Form the time of FoxTrax.
My Mom has had endzone seats for the Patriots for 35 years. Occasionally (once a season or so) a field goal/PAT will go wide enough to miss the net behind the goalposts and land in the stands. Whenever that happens, Security comes and confiscates the ball from the fan who caught it (I don’t think they are given anything in return, but I could be wrong).
A couple of years ago the field guys who raise the net for the FG attempt were too late, and the ball sailed through the posts and into my Mom’s lap. She quickly tucked it under her coat. Security came by and knew the ball was in her area, but not who had it. So they kept saying, “Who has the ball? You need to give us the ball or you will be escorted out of the game.” But everyone in my Mom’s section looked at them blankly – “What ball? What are you talking about? We didn’t see a ball.” They finally gave up and walked away.
After they were gone, the guys who had covered for Mom wanted to see the ball, but she refused to take it out of her coat. The following preseason she took the ball to a public practice event and had it signed by several players including Drew Bledsoe, who was impressed that she had a actual game ball for him to sign (briefly she was afraid he was going to take it away from her).
Baseballs are not always for the fans to keep. In college ball (at least here in Big Ten country), if you get a ball you have to give it back. They even send a player out to talk you into doing it.
I’ve heard legends about fans smuggling in a cheap football, then if they catch one they hide the game ball and toss the cheapie around. By the time it gets down to the field, they’ve deflated the game ball and stuffed it in their clothes.
Hockey pucks are fair game wherever I’ve seen it played.
From the Offical Rules of MLB
5.11
After the ball is dead, play shall be resumed when the pitcher takes his place on the pitcher’s plate with a new ball or the same ball in his possession and the plate umpire calls “Play.”
(Emphasis mine)
Balls in Play
So the rules allow balls to be reused. Unforunately, my first hand evidence for balls being reused comes from being in the stands, which is awfully hard to cite on the web. 
Balls hit in MLB games are not always replaced. The general rule is that they will be replaced if they have been hit such that they came in contact with the dirt portions of the field, or, upon inspection, if they have hit some other item (Wrigley brick, e.g.) which scuffs them. But contact with the grass won’t necessitate removal from play.
The golfer can still give his ball to someone on the side of the fairway… He or she would just be penalized 2 strokes.
From the USGA:
1-2 Exerting Influence on Ball
A player or caddie must not take any action to influence the position or the movement of a ball except in accordance with the Rules.
(Removal of movable obstruction — see Rule 24-1.)
Penalty for Breach of Rule 1-2:
Match play — Loss of hole; Stroke play — Two strokes.
Note: In the case of a serious breach of Rule 1-2, the Committee may impose a penalty of disqualification.
Tennis balls are returned. They use the same balls for 7 games then they are changed. At the end of the match the players can(and do) hit the balls in to the stands.
The local junior college allows fans to keep any baseballs that leave the field of play. They can afford it – balls escape so rarely that it’s not a big cost factor.
The same college has a nationally-ranked basketball team, and I do arena announcing at the games. Balls occasionally go into the stands. Several years ago, during the annual homecoming game, the team honored the last living member of the very first basketball team (1941). The guy was sitting right behind me, practically at courtside, and a pass got batted off the court. The ball whizzed past my ear and right into the old-timer’s lap. He caught the ball perfectly. The coach jumped up and yelled, “Let 'im keep it!” The scorer tossed in the spare ball and the team autographed the game ball after the game (we won.) It was pretty cool.
Minor nitpick: if the ball is damaged, on that hole, then it can be replaced. Remember the old balata skinned balls and how easily a thin shot would inflict a “smiley face” on the surface? Of course touring pros hardly ever have that bad of a mishit.
At the 1978 TPC, Jack Nicklaus threw his winning ball into the milling throng…right at me. Of course being the bonehead that I am I tried to catch it one handed, and it hit my palm and bounced away (some girl got it). He was my idol too, that’s what hurts. Yeah yeah if I catch it I work like crazy to become a touring pro knowing that it is now my destiny…
In my own defense I did include the phrase “or is otherwise unplayable.”
I’m new to posting, but I have been watching y’all for a while. I know how pedantic Dopers can be. 