According to this site which I got most of my information from.
You currently get some profile indicators, as well as other bonuses.
First bonus for getting that you are a great teammate is this during champion select:
Same thing but for being an honorable opponent
These also show up during the loading screen
The popup once you are alerted that you have received one of these alerts
Right now the exact amount of times you need to be honored is unknown. This is more then likely due to the honor votes being weighted in the background. Someone who just their friends votes count less then random people you play in matchmaking.
Right now the only way to loose honor is if you are reported to the Tribunal and found guilty of what ever infraction you were reported for. This resets all your honor to zero.