Nasus main reporting in. I've played against all of these champs many times and I have some opinions. Some good info in here, but also a lot of stuff I disagree with. First let me point out some of the ones that people have pointed out that are either wrong or exaggerated.
The most you're going to do is annoy him. With jungler help you can maybe push him out of lane, and drop t1 tower. That's the best case scenario. Once he gets SV he can just kill you, and while you can slow his stacking you can't stop it or kill him. He will be more useful at splitpushing AND at teamfighting than you once he spikes. I see people make this mistake all the time, Teemo players are free wins.
Huge mistake. Illaoi thrives off of winning trades and shutting down all ins. Nasus doesn't want to fight you. He wants to farm. And he can get stacks off your tentacles. Nasus counters the shit out of Illaoi.
Not enough damage. Nasus will start Doran's and get Cowl first back then farm and ignore you.
Wither DESTROYS Jax. If you get ganks and get fed it's possible to bully Nasus severely, true. But if it's a 0/0 kinda lane you are not gonna win that by yourself. And Nasus is one of the few champs that outscales you and can shut down your splitpush. Don't do it.
You will control the lane, for sure. But unless you're 100% sure you can outplay and kill your opponent before he completes an armor item, don't do it. Riven goes splat to stacked Q's, she won't be able to duel a Nasus with IBG even if she has a CS lead.
So you harass with autos. That draws minion aggro and pushes the lane. Then you get withered, ganked and die. If his jungler is afk this can work, but I don't recommend it.
Nasus cannot fight Olaf before 11. But again, Nasus doesn't want to fight. This will be an uncomfortable lane for him because of how safe he has to play, but Olaf is not going to shut down a Nasus.
Not a fun lane, but not scary. You have to play extremely defensively, but you'll still hit 300-350 by 20 minutes, so Garen doesn't shut you down at all.
In terms of champs that can make it impossible for Nasus to farm and ruin his day?

Panth and Fiora should be pretty obvious. Kayle is one that a lot of people overlook, but she's a hard counter. Nasus might as well go farm jungle. I will usually dodge rather than lane against these three champs.

These are all pretty rough, just not AS bad as the three above.

Trundle is a weird one because he doesn't shut down Nasus as hard as others, but he can duel Nasus pretty much all game if he has ult. Trundle shuts down Nasus splitpush single handed, there's no way to get past this guy if he just sits top lane. If you're pretty sure your team can win 5v5 (you have a comp that can kite Nasus, for example), Trundle is an incredibly solid pick.