I keep seeing tanks and being able to walk right up to them seems like a huge advantage when the gunner is looking somewhere else... but I'm not sure if there's an unlock that I still need to get that would give me something to completely disable an armored vehicle at close range.
Shooting at it with a weapon is pointless. A rocket launcher would work but takes 2 or 3 hits.
There are 3 ways to take out a tank from close range.
The first is as Support, to use your C4's. 3 of them will destroy any armoured vehicle from full health.
The second is as Engineer, use the repair tool on the enemy vehicle to damage it fairly quickly.
The third is also as engineer, and just involves planting anti tank mines around it and hoping it drives onto one of them.
I'd vote for Mr Smooth's reply as the answer for this one. But here's a corollary tactic you can use if the driver suddenly figures out you're there.
Sometimes a driver will jump out of the vehicle and try to attack you directly. This can happen if they see you running up with a kit that carries C4 or anti-tank mines, or if they hear the repair tool and see their vehicle's health going down.
I saw this behavior numerous times in Bad Company 2, where I was constantly C4ing tanks. I also saw it when I played engineer, which was less frequent. I haven't unlocked C4 yet in BF3 but I'm pretty confident we'll see the same behavior in BF3.
Depending on the situation, it's not totally stupid to jump out and confront your saboteur directly. For example, if you see the C4 coming too late you might have as little as 1-2 seconds before the attacker clears the blast radius and detonates. You can only stop that by eliminating the attacker.
When you hear the engine stop it means the driver has jumped out. That's your cue to try jumping in. (You won't always be able to: I need to confirm this, but I believe the engine stops even if a gunner is still in the vehicle.) Once you're in, start driving immediately. At worst you get away with the tank, and by driving immediately you sometimes get lucky and roll over the former driver.
So keep your options open. Disabling a heavy vehicle with a sneak attack is fun, but stealing the vehicle and pancaking its former driver is way, way more fun.
I did a bit of research and I think I have found the answer:
Use the Repair Tool
It turns out the repair tool is also an un-repair tool when used on enemy vehicles.
Besides what Mr Smooth and Drew G have said, you can drop mines directly behind (or next to) the tank, and then throw a grenade at them, which leads to a nice big explosion.
I haven't had the chance to test exact numbers, but this will cause major damage to the tank if it stays in place.
