Azir shares a weakness with Orianna.
Let them use Q, then get out of the way. Both of them are screwed during those few seconds Q is on cooldown. The majority of their damage is tied to a short radius around their objects.
Of course Azir has his wall to try and deal with you if you avoid the soldiers. But there's a good way to counter that, too. Blinks, specifically ones like Talon's and Katarina's, are very hard for Azir to handle if you can get him to ult first (or if you have the burst to kill him outright). And presumably most teams have a frontline tank that can try and get in Azir's face and force him to ult, leaving the way clear for an assassin to eat his face.
If he ults you off and you don't have a backup blink to get over the wall, don't waste your time walking around it. Back off and try again later.
His poke and escape are tied together; using one of them locks the other for a short time, because both require using Q. The instant he pokes you using Q, punish him for it if at all possible.
The main thing in lane is to at least avoid letting him hit you a soldier without even using Q. If you get that close, he can attack, Q, attack again, and maybe even get a third attack out of it.