Golf is the only sport where TV viewers act as rules officials. If a fan sees a violation and it turns out to be true, a player must be penalized. How replay comes into play in other sports and who can invoke it:
Officials can use video evidence to decide certain plays when coaches (not fans) challenge them. A replay official automatically reviews turnovers, scoring plays and some other loose-ball situations and can request the on-field officials review evidence.
Umpires use judgment for all calls, but can use video on home runs to determine whether the ball fair or foul, or if it cleared the fence.
Referees can check courtside monitors to determine if a basket was a 3-pointer or not, but no calls may be directly influenced by fans.
NHL officials monitor games and initiate replays from a “war room” in Toronto to determine if a goal has been scored or not. There is no outside mechanism for a TV crew or viewer to alert someone of a rules violation.
Sources: NBA, MLB, NHL officials; NFL.com, NHL.com