Current Balance
This will be the first time posting on the forums as up until recently I haven't really had to much to complain about balance wise with the game. With a lot of the new changes and champs having been released a lot of things now seem out of whack to me and while they are being touched on in other posts, I thought I would share my opinion of everything in one post rather than jumping between many different ones.
Now to the point. The recent champions released.

While I'm generally ok with Gnar, I do feel there is one glaring problem with him and that is the %max HP damage attached to his mini Gnar W. The problem I see with this is that in big Gnar, he is a team fight monster and pretty op, which is completely fine by me. I like the fact that you don't want any part of big Gnar but the problem for me is, why when he has this, is he not only extremely safe in lane but also a lane bully? The main reason he is such, is not due to the fact that he is ranged but almost exclusively due to his W passive.
So to not make him so obnoxious in lane, I believe this damage should be removed and replaced with a buff to the move speed that he already gets from it. My reasoning behind this is, the %max HP damage isn't used all that much later game and while it is a nerf to his early game and lane, meaning more melee champs can be picked against him. It is essentially a small buff to his later game as with more speed from the W 3rd attack, there will be times that this can help him get amidst the other team a little easier.

Now I touched on Gnar being obnoxious in lane and I believe Azir to be far more obnoxious and not just in lane.
and
where both changed (one more than the other) for being able to put out harass without any chance to reply due to there silence. So why is Azirs entire kit built around this?
He can zone and poke without being in risk of taking much damage, if any, and if you take an all in champ against him, not only does he have his ult, he has mobility on top of it. Now I know it wont be changed as it would require a complete rework and the idea behind his kit is a cool one, so instead I'll just propose so ideas to reduce obnoxiousness.
First one would be to reduce the range to which he is linked with his soldiers, pretty simple. Secondly this is the one that I prefer, anytime he uses is soldiers to attack it costs him mana, additional soldiers cost additional mana.
I'm not sure if adding both of these would be overkill but some testing would surely find it out.

......... What were you thinking................ ............. Since I doubt you'll remove that absolutely horrendous passive, lets see what we can to to not make her so utterly revolting as a champ.
First things first, if your going to give her that much mobility and utility you surely can't make her a hyper carry........ and you did...... To fix that put a cap on the amount of spears much like Twiches expunge and remove or massively reduce that absurd %max HP of the W.
If reducing the damage is not your sort of thing, then reduce her AA range, add a cap to the spears against minions and make slows affect her jumping distance.

Following the recent damage nerfs, not much can be said until a bit more time is given to see the effects but one thing for sure has to go. That is when she is burrowed DO NOT LET the rest of her team see the footsteps of the enemy team. There is literally no reason why this should happen what so every, it doesn't even make sense as to why this would happen and it such an unfair adavantage to the team that has Rek'Sai.
It would bring about more teamwork and communication if only the Rek'Sai player could see it. If the damage nerfs and this are not enough, you could always add a cap of 10 or so to the amount of tunnels she can have at any given time.
Ok now to older champs

Lets be honest here, mana regen nerfs brought him to the limelight and the attack speed nerf was next to worthless for his main problems which are wave clear, unlimited harass and itemization.
Wave clear can be reduced by increasing the reduction of damage to his Q when passing through minions.
Unlimited harass can be reducd with having his Q cost 70 energy at all ranks.
Itemization (ArmPen I'm looking at you) can be changed with either reducing base damages and increase scaling means more flat +AD damage needs to be add to the ArmPen builds or removal of the %AD bonus on his W.
While I'm not exactly sure as to what should be done, as I feel that overall he isn't to far from being balanced.
I believe the best course of action would be the 70 energy cost to Q at all ranks and the removal of the bonus AD. As it limits his Q spam a little in lane, it still allows him to cast all abilities at any point in the game so not really affecting him in other areas. The removal of the bonus AD means that he may have to choose more between ArmPen or flat AD depending on the effect this has and the way the game is going.

While I liked assassin Ahri I'll reserve judgement until I have played her more.
As for
I'm not exactly sure what to do aside from energy costs need to be more meaningful but preferably I'd just rather other junglers got buffs.
Now I'll touch on the mana regen nerfs. While justified some champs mana costs have been left behind and need to be addressed eg. 
needs its cost and % regen brought in line with
to make it the defensive opinion and not just make
to give more build paths and item diversity.