Okay, so first of all, you could deactivate this, just like you can Smash Balls. In fact, that's how you'd activate the Tag-Team Combos (and hence TT-FSs)--with Smash Balls.
But first, let's lay down some ground rules. There's a new game mode in VS. called "Tag-Team", in which you select three characters per person (it's still a 4-person game, at most). You can do teams, but that won't affect your potential Tag-Team Combos/Final Smashes. When you pick your characters, you assign each to either UP, LEFT and RIGHT, or DOWN on the D-pad, replacing your taunts with character switch.
So all you can do at first is switch characters. Switching to the character already in play just activates their "Tag Special", which is just one of their regular specials, but which special is used differs from character to character (Having Kirby use Inhale in a combo would be rather useless, after all). Switching to another character after a tag special just switches, it doesn't activate the Tag-Team Combo. For that, you need to have a Smash Ball.
Once you break open a Smash Ball, you can still just press B to use your character's FS, or you can use the Tag-Team Combo. Note that even if you don't finish the combo (if you just do Mario>Link, then stop, you just get Fireball>Boomerang, then nothing), you still use up your ability to use your Final Smash. If you do finish it, however, you get the full three part combo (Mario>Link>Meta-Knight = Fireball>Boomerang>Mach Tornado with no ending lag), which always features a "super" version of the last character used's Tag Special, in this case Meta-Knight does a full-powered Mach Tornado, but after it ends and he's in the air, he can immediately double-jump away.
Of course, if you were using Mario>Link>Luigi, you'd get Fireball>Boomerang>Cyclone Bros.
(Luigi pounds Mario down and jumps on his head. They both spin whacking the enemy time and time again with the hammer. This is Mario and Luigi's most powerful, non-elemental Bros. Attack from Mario & Luigi: Superstar Saga)
...Which would also use up your Smash Charge. Note that no matter where Link is used in the example above, Cyclone Bros. is still used, as long as both Mario and Luigi's directions are tapped. The only way that would not happen is if you pressed Mario>Luigi>Link--to get a Tag-Team Final Smash, one of the involved characters must go last. If this wasn't the case, you could get an extra attack in by using Fireball>Green Missile>Smashed Boomerang>Cyclone Bros.
---------------------------------------------------------------------------
Got all that so far? Okay, now it gets complicated.
---------------------------------------------------------------------------
Team Battle: The Team Healer requires some level of coordination, but I think this takes it to the next level.
During a team battle, the only way to do Tag-Team Final Smashes is to coordinate button presses with your teammate, who is also playing as a character you can 'Combo with (i.e. you are Mario, he is Luigi). All you have to do is be next to each other (no farther apart than a dodge roll would carry you on the ground, and no farther apart in the air than a Melee-style air dodge would take you) and press B at the same time, and the TTFS goes off without the inclusion of other character's Tag Specials.
Note that in a Tag-Team Mode team battle, there's no special rule like this--Tag Combos and FSs are activated normally.