I play AP yi, a lot. I've ran through bronze 5 to bronze 2 in a week, with only 4 losses. Roughly 20 games all as AP yi. That's not a lot by any stretch, and the people I'm playing against are awful. It's not that rare for me to break 1 kill/2 minutes. Most of my deaths are really dumb moves on my part...but this is all irrelevant.
I've noticed a few counters as I've been playing, and since it seems to be a really common complaint to all the people I play both with and against, I figured I'd shed some of my lol1kelo insight onto this.
AP Yi is trash at the start.
He can CS, He can poke, but he damn sure can't kill and he can't even escape very well. Everytime he qs, if you were in range when he started it DON'T RUN. If it hit you, it's already hit you once he uses it. Poke him back, punish him for every.single.cs he gets. I don't mean just throw your spammy attack, Go all in on him every single time it's safe.(no jungler nearby, and you aren't under 50% hp) His heal is awful early game, he can't sustain meditating more than 3 or 4 times in lane without blue, and he can't help his jungle gank. PUNISH HIM FOR ALL OF THESE THINGS.
AP Yi is trash in team fights, but he's waiting around the corner to ace you.
This is the biggest thing that I carry my team on. Oh look, you guys are 5v4...and you're all at full hp except that support at 30% hp and still standing next to you... Flash Ulti Q Q Q Q Q "PENTAKILL". This is the biggest thing, if you're under 40% hp, and Yi is not dead, GET AWAY FROM EVERYONE. I don't care the scenario, I can ace any team if they have one person under 40%. My Q is faster than your stun, and I'm untargetable. This is why people bring Yi to the game, If you don't let him get his resets rolling he's an AP mid who has 600-800 damage+lich bane auto attacks every 7 seconds, and that's literally it.
AP Yi doesn't do anything without farm.
Shut him down, moreso than any other 'snowball' character, a fed yi can ace a team 1v5 if the adc isn't buying tanky items. I've done it numerous times, 1 DFG>Q>Auto attack into a Q on support into a Q on the APC into a q on the top/jungler...This is how I snagged my first 1v5 pentakill. If you let yi free farm mid, he will dominate mid to late game. Don't give him kills, Shut him down in lane, it's not hard, his early game heal barely even outheals ignites damage on its own, it's not til he buys that needlessly large rod does he even become a threat. Most Yis, myself included, won't back until the first needlessly large rod is affordable. Punish them for this, it means no wards and no survival items early on, just the silly little doran's ring.
AP Yi is not OP, he has many of ways to counter him, and even ways to outright make him useless. I'm not the best yi, and there are definitely ways for Yi to counteract all of his downsides, but ultimately Yi is the most gimmicky mid currently. His gimmick is scary, but it's very easy to counteract if you just think through his pros and cons, and simply play smart against him.
No champion is unstoppable, every champion has a weak point. Yi has many more than just this, but watch what he buys, play smart against him, remember that he can't trade very often(at level 1 once every 15 seconds), and that he can easily be beat. Best of luck.