Do Leavers gain rewards?

If you see "Leaver" on the end of match summary screen, that summoner received no IP for the match.

However, just having left the match early is not enough - League of Legends has a 5-minute grace period before a summoner is counted as a leaver (and it gets reset if they rejoin). I once received a "win" from a game I did not finish (powersurge) because the enemy team surrendered before I had been gone for 5 minutes.

Neither of these affects the fact that you can report someone as "Leaver/AFK" via the Report Player button. While these summoners might still earn ill-gotten IP, they're also likely to get banned or suspended from their behavior (rendering the point moot).

A players gets the 'Leaver' mark by leaving the game for whatever reason (DC, Rage, etc..), but only on those cases when the leave overcomes a time threshold.

Clearly as James stated, a tagged player at the end of the match will not receive any xp/ip rewards by fact.

However, say the game lasted 40 minutes, and you had a DC'ed teammate who left at the 35 minute mark, he will not receive the tag if the threshold is greater than 5 minutes.

They gain nothing but the leaver flag which can also be reported on and further action can be taken against the person. People who leave cause their power fails or some other reason are generally OK.. People who rage in the chat logs and quit or idle afk are generally reported and get reprimanded.