Half of your list are tanks, and tanks are the weakest role in the jungle right now.
Nocturne isn't strong early. Until he has a few points in his Q and his ult up his ganks are weak, and his damage in 1v1s is meh at best. The only hope Nocturne has of winning against a Yi is if he can get Yi to Q into his spellshield. A smart Yi will enter the duel with the Q, or just save it for after Nocturne has used his shield, in which case Yi beats Nocturne.
Volibear isn't strong early. He requires a significant health investment to make his execute deal any considerable damage, and the only way to make him strong early when it comes to dueling is to give up early health in favor of a Devourer's enchant to supplement his auto attack damage. But given that he lacks significant sustain in the jungle outside of his passive, attempting to counter jungle a Yi with him is asking for trouble because a Yi, even pre 6, would have no issues tearing a half health Volibear to pieces. And if he builds a Devourer instead of Juggernaut, Yi can shred him from full health without much more trouble. Until he hits level 6, Volibear's single target damage is meh at best. So if you're level 6, so is Yi. Skipped his touchy early game.
Amumu certainly isn't strong early in dueling situations. He lacks mana to keep his tears up or spam his tantrum, and he isn't doing a whole lot of damage with them early on in the first place. By the time Amumu's in a position to actually be a threat 1v1, you're in the mid game.
Jarvan isn't going to keep up with a fleeing Yi, even popping his slow on him, and he has no way to trap him pre6 to keep him within range for a beating. Even then, Yi's Q is up faster than Jarvan's spear, so you aren't going to win that trade either.
Kha'Zix isn't strong in 1v1 situation until after he has a few more points in his Q, in which time Yi's been running around in his own jungle maxing his Q, which deals more damage and is up not quite as much as Kha's but still insanely fast, and has also been clearing his jungle quicker so he has a gold and level advantage. In fact, in order to beat out a Yi early on, a Kha'Zix has to sacrifice early ganking potential by evolving his Q first at level 6. Which means Yi is at level 6 as well, which once more means that Yi has been able to farm to 6, proving my point.
Lee Sin I'll give you. The extra durability, movement options, and the % health execute will let him beat out a Yi if the Yi isn't smart enough to tank most of Lee Sin's burst with meditate.
Mantheon might defeat a Yi, but only if he gets the drop on him. And even then, Yi can probably still run away or vision hop over a wall with a ward and a Q to a minion camp. But yeah, the Speedo'd Wonder could probably pull it off.
Rammus isn't going to counter jungle a Yi successfully. His own clear is abysmal during the early game, and he is not going to deal enough damage, nor mitigate enough damage, from Yi's attacks to let him survive. Even if you rush a Thornmail, Yi's true damage doesn't care about your armor and he's going to be dealing more damage to you than you can do back to him even with your W active. Rammus doesn't gain the potential to counterjungle until after maxing E, putting points into W, and building at least a Sunfire Cape on top of Juggernaut. The early phase of the game is over, Yi is fine.
Rengar? You pop your initial burst on him, which is pretty weak all things considered early on before building Warrior or Devourer, and then he just hits meditate while you keep trying to auto attack, and once meditate is down his Q and his E are back up. You lose.
Rek'Sai's damage early is terrible. She needs Devourer's before her damage can even begin to matter, and Hydra before she can safely start trying to 1v1 people. By which point, again, Yi is no longer in his early game either. My point once again proven.
Shyvana? Quite possibly. If Yi is low, and if he doesn't meditate through the duration of your W to reduce how much damage you can deal to him after it's finished.
Skarner gets chopped to pieces. Don't get me wrong, I love Skarner and all, but he has to powerfarm for a Devourer's before he can even try to be relevant, and his ganks are weak pre 6 because his passive stun is meh at best early on. Once again, Yi safely emerges from the early game phase.
Udyr? Maybe, depending on your skill order. Chances are you have to reduce your damage output by leveling turtle form first, or gamble on increasing damage output by leveling tiger form first. In either case, you're taking a risk even trying.
Warwick isn't counterjungling Yi. Yi is counterjungling Warwick.
Wukong might be able to do something, if you can get him to waste his Q on the clone. But once your E Q combo is blown, and his armor returns to normal while your attack speed drops, Yi takes the win because his autos are hitting faster, and every fifth is hitting harder. Wukong needs his ult to put the finishing touches on Yi, meaning that once again Yi survives pre6 without an issue.
Xin Xiao tries to go ham on Yi. Yi presses W. Xin wastes his burst, Yi returns the favor in spades. Until Xin has his ult, his single target burst isn't high enough to counter how much Yi just flat doesn't care.
None of the champions that you've listed off can shut a Yi down in the early game. He simply deals too much damage through his Q, E, passive, and natural attack speed, reduces too much through his W, for him to be stopped by any one champion specifically in the early game race for 6. And keep in mind that during the time you're wasting traipsing around his jungle trying to find camps that he hasn't already cleared yet, or to try and find him, you aren't farming while he still is. So the more you try to counterjungle someone who's clear speeds are pretty high, the more advantage you're giving him in gold and xp because you aren't getting any.
My point stands. By taking Yi into the jungle, he doesn't have to worry about his early game, and by trying to 1v1 him if you do manage to catch him, he's more than likely going to take the kill, taunt on your body, and spam "ez" in the all chat.