Are you allowed to play zone defense in the NBA?

Zone defense is not explicitly prohibited, but there is a defensive three-second penalty occurring when a defender is inside the lane without guarding an offensive player.

From the NBA Rules:

Section VIII-Defensive Three-Second Rule

a. The count starts when the offensive team is in control of the ball in the front-court.
b. Any defensive player, who is positioned in the 16-foot lane or the area extending 4 feet past the lane endline, must be actively guarding an opponent within three seconds. Actively guarding means being within arms length of an offensive player and in a guarding position.
c. Any defensive player may play any offensive player. The defenders may double-team any player.
d. The defensive three-second count is suspended when: (1) there is a field goal attempt, (2) there is a loss of team control, (3) the defender is actively guarding an opponent or (4) the defender completely clears the 16-foot lane.
e. If the defender is guarding the player with the ball, he may be located in the 16-foot lane. This defender is not required to be in an actively guarding/arms dis-tance position. If another defender actively guards the player with the ball, the original defender must actively guard an opponent or exit the 16-foot lane. Once the offensive player passes the ball, the defender must actively guard an opponent or exit the 16-foot lane.
PENALTY: A technical foul shall be assessed. The offensive team retains possession at the free throw line extended. The shot clock shall remain the same as when play was interrupted or reset to 14 seconds, whichever is greater.

This means that a college-style zone is not possible, but a more limited zone can be done. Usually this is only a change of pace and not a base defense.

According to Wikipedia, the NBA allowed zone defenses starting in the 2001-2002 season:

The NBA currently permits the use of zones; however, teams generally do not use them as a primary defensive strategy and no zone defense may feature an unguarded defender inside the free-throw lane (a violation of that results in a defensive three-second violation, which is a technical foul).

Yes, it's allowed, even if it's not often used.

For example in the last NBA finals, the Dallas Mavericks often used zone defense against the Miami Heat. It was one of the reasons for the victory of the Mavericks along with bad statistics of Lebron James in the finals (but not the only one).