Why did Bourne fire a shot in the air?

It's a classical "I pretend to have to fired a shot, so you think I have revealed myself involuntarily, so you make your move, and then I get you because I'm ready" move.

The basic idea is a confusion. Bourne fires a shot, which would normally reveal his position. But, the birds are scattered along a wide area, so they create a mass distraction. The Professor now has some hint (from the direction of the shot's sound) where Bourne is, and tries to use it to his advantage, by moving in that direction.

This would work for "The Professor" just fine, had Bourne been distracted, i.e. if he fired a shot accidentally, or as a defense against some animal. Since the shot was staged, Bourne was ready and spotted him.

Had "The Professor" not moved, the birds would be useless to Bourne, but he instinctively moved and it cost him.

While the reason behind this shot (and any other shot just like this) could be, just like Vedran suggested, a trick to get the other person to believe that you for some reason had to fire your gun at something (or someone) else and must therefore be distracted making it a perfect opportunity for the other person to take off; the reason could also be to make the person believe that you know where he his but just happened to miss the shot. In either case, the person being hunted is unlikely to stay in his current position after hearing a gun shot. While the actual, specific reason varies, the basic idea is to trick the other guy to reveal himself by making a run for it.

My view is that the birds when they came back to land in the field would not have landed anywhere near the prof therefore giving Bourne an idea of where in the field he was hiding. The fact that the shot made him move was a bonus.

People you miss something here: The Professor is armed with a sniper rifle (and maybe other weapon) while a scattergun is all Bourne got.

Bourne knows for sure there is no chance to win a long range with a scattergun. So his strategy is quite clear: get close to The Professor, make the battle a short range fight.

He shoots the tank to make smoke his cover that enables him approaching. Later the shot in the field does the same trick: flock (and noise) covers him and discracts/confuses the sniper.

The Professor's reaction proves it: He switches to pistol after moving even though he doesn't spot Bourne. He knows a sniper rifle is useless when you lose sight of the enemy in an environment of poor visibilty.

Jason Bourne has "Lateral Thought," like most trained assassins. Though the professor lacks counter thoughts to Bourne's lateral thinking, the scene illustrates how to unconventionally solve and or create seemingly dire predicaments.

I say dire because it plays into the professor's blunder. In an intense situation like this, his flight or fight kicked in when it shouldn't have.

Bourne doesn't know how many assassins are in the field (as he asks the professor how many of them there are). Scattering the birds accomplishes several things. Bourne might be under the assumption that there is more than one killer, fires a shot to indicate that one of them was hit or spotted (if they're separated), to emotionally compromise their future decisions. Had the professor correctly decoded Bourne's move, I doubt that he would have left his position. That and, the gun that Bourne was using is distinguishable, and further feeds valuable information to the Professor.

Next, Bourne has a short range weapon. Scattering the birds may not only confuse the remaining assassins, but allow for him to get closer to them. If he had not scattered the birds first, then ran through the field, the birds would have given him his precise location.

My biggest problem with this is that the professor seems to lack lateral thinking. Of course it would be harder to showcase Bourne's character or hold on solving problems if their match was full of counter moves. Hard to see that he's good at solving problems if he keeps creating new ones. But it would have been interesting to see another versus with another assassin later on that could counter Bourne's moves.