To quote the Wiki,
"The on-fire meter is an indication to the player as to how well they are performing with a hero. The meter starts off as blue, and turns to red as they carry out actions that adds to the meter. When "on fire," the hero's character portrait gets a flaming halo. The meter can get more 'fiery' through actions such as scoring kills, completing objectives, and denying objectives to the other team. The enemy team will also know if a hero is "on fire" as a mechanic to alert other players of enemy player skill, allowing them to adjust their tactics".
On-fire status is also shown on the team summary screen shown during matches.
That said, it's basically just an indicator of skill, displaying how much of a threat you are to the opposing team, which may be useful if you want to cripple the enemy team by killing their MVP.