With ignite I can get first blood and secure kills MUCH easier, but I find its basically useless on him late game because the range you have to use it is counter-productive to using his WQ combo. However, I'm not sure I like tele enough to take it instead. I'm considering barrier, but I like the map pressure with tele more than a small shield. Help?
Azir isn't really meant to be a kill laner. He can pull it off sure, but that's his secondary style. Azir is much better as an elusive poke style champion. I find that taking exhaust is much better on him because alot of times that 1 extra auto attack from one of your soldiers will deal more damage than ignite would. That and the fact that exhaust can be used in a defensive/offensive manner makes it worthwhile. Taking exhaust just makes it harder to reach the already difficult to reach Azir.
On Azir I recommend 
or 
in most cases. Barrier against burst damage such as 
, Heal against sustained damage and poke like 
Xerath could be a candidate for taking either Summoner, your choice.
Exhaust can be considered too against really poor match ups, say you get counter picked or something.
I don't light ignite on champions like like Azir, Xerath is another prime example, if you're within ignite range on Xerath you're doing it wrong. Azir is a bit of a different story, but just the same, I don't think it's an optimal spell for him either.
I would take teleport for sure, the presence is impacting and Azir is the best back door in the game now since your W hits a great amount of damage on towers, your ganks are monumental since you just slide into lane with teleport and use your multi, guaranteed that everyone on the lane uses flash or end up dying.
Not to mention that Azir is a farming champ, his build is slow sine you don't get your impact on the game till you finish building Nashor's Tooth, till then you won't be able to show your true presence to other lanes so you may use teleport to maintain a good farm by staying in lane more often.
Ignite is nice in early game but azir isn't really played for the early game. He's more of a late game champ with unreal poke. Heal gives you the best utility come late game but barrier will help you survive hard all ins early to mid game.
On long range pokier champions (
, 
, etc.) , I generally take barrier + 
/ . Barrier scales pretty well and really helps with unavoidable damage like Caitlyn's R or Malzahar's E.
I take cleanse or barrier with flash. I will never take teleport on a mid laner because I simply don't like the spell and I just don't like ignite on Azir. Exhaust is fine as well.
Depends on your opponent. If your lane opponent is a late-game powerhouse, bring Ignite to shut them down early. If your lane opponent is very poke heavy, pick Heal to get more lane sustain. If your lane opponent has a lot of burst, take Barrier and pop it immediately when you see them moving in for a big all-in. If your lane opponent is a stunner (like Irelia, Annie, TF, etc), you can bring Cleanse to get away, burning the enemy's CC in the process. If your lane opponent isn't very threatening and you think you will be able to be influential in other lanes early with TP, bring it.
