Depends on how you view tier 3. If you mean "Can one beat the other in a 1v1 duel?" then no, the barbarian or cavalier isn't likely to beat the dread necro on most days. But that's not really what JaronK meant by his tiers.
If you use it to mean "Can one competently fill his primary role in the party and do 1-2 other things decently?" then I think a barbarian needs some ways to keep from being sidelined by failed saving throws, much like the warblade, as well as ways of dealing large damage when it can't full attack (either from using PA+Charging feats in 3.5, or Vital Strike in PF) to hit T3. In other words, an optimized barbarian is T3, but most players who don't pick their feats, PrC, and race particularly well will probably end up at T4. Thankfully, barbarians are pretty easy to optimize.
For the Cavalier, I think it's probably T4 without some optimization. The PF paladin is definitely T3, able to fill its role as a tank, and also work as a party face, backup party healer+status effects guy, and deal some solid damage with smite evil. The Cavalier can also work as a tank and as a party face, but its buffing to the party is pretty minor, and its damage is probably behind a paladin smiting evil. With the right order (Order of the Shield is terrific for tanking, Order of the Sword is a significant damage boost, Order of the Cockatrice can do the Inspire Movement of the marshal at will), and the right chain of feats, the cavalier can be T3.
If you wanted to make the class a baseline T3 (so with additional optimization you're effectively T3++, since T2 and T1 means you're breaking the game instead of just being better at what you do), I'd do the following things:
1. Increase mobility: As mentioned, the addition of some more mobile mounts to the animal companion list is a good idea. Additionally, the cavalier should probably get the fighter's ability to move easily in med/heavy armor.
2. Increase the relevance of the cavalier's buffs so that the party actually cares. For example, if every party member who hit the opponent in melee gained the cavalier's Challenge damage on attacks, that would probably boost the cavalier to one of the top buffers for most parties - it would be like having Dragonfire Inspiration from a bard.
3. Give it cha to saves like the paladin, to shore up its main weakness.