Could a Barbarian with Totemic Attunement (eagle) use grapple to perform the iconic move by dragging the enemy straight up as far as the fly speed allows and ending the turn in the air? If so, who would take fall damage?
Yes, you can do this. No you won't get very far, and I'd definitely rule that both of you take the falling damage.
First, yes you are allowed to move a target you are grabbing. So you can feasibly grab them with your action, then use your movement to fly with them.
However, since you are moving with a grabbed target, your movement speed is halved, so you only get 15 feet up. This means that the falling damage is at most 2d6 damage (though some DMs may rule it to be 1d6, I'd be inclined to) and you'll be prone..
Ultimately, it looks like you both fall. I'd definitely rule that since you're falling the grab is up. And, since there is not a direct way to negate falling damage, unless your caster friend is shooting feather fall as a reaction, you're both taking damage here.
If I may editorialize here: this is not a good strategy. It consumes your action with the grab which is then most likely broken, and only does 1d6, maybe 2d6 damage, which is what your weapon does, and it's likely less reliable since it relies on an opposed check rather than an attack against AC. Stick to what you're good at: hitting things with metal attached to sticks.
You can give fall damage by jumping in the air while grapling. You both would recieve fall damage and be prone unless an ability like slow fall or feather fall says otherwise.
However, the barbarian is not best way to achieve your goal. The best way to pull off a pile driver like you are describing would be with a Monk/Druid. You only need to multiclass one level of Druid at level 2 in order to get access to helpful spells like longstrider, jump, and Guidance. The Monk gives you useful features like Slow Fall(reduce fall damage), Step of the Wind (use dash as a bonus action), and Extra Attack. Be a human, so you can get a bonus feat as a level one monk (if your DM allows). I would get the Athlete feat to start with and grab the Mobile feat at level 5 (remember you have to wait cause of the one level of multiclass into druid at level 2)
Here is the full pile driver combo,you grapple a target and then make a standing high jump. At the top of your jump, you hurl them to the ground, deal damage to them from the fall, knock them prone with no save, and then just re-grapple them in the same turn. This works by using the Jump spell to triple your jump distance, Step of the Wind to double it, and then stacking move speed bonuses so you can get to maximum height even with half move speed. Here's how it works assuming a base speed of 50 (30 + Longstrider 10 + unarmored Monk 10) and assuming you have Extra Attack.
Start next to your opponent and grapple them with the attack action (attack #1).
Take the Step of the Wind bonus action to Dash and to double your jump distance. Your move speed is now 50 for the turn (50 base / 2 grappled * 2 Dash).
Take a 5 foot step (45 remaining) to trigger Athletic.
LEAP FOR THE HEAVENS. Your jump height is massive at 36 (3 base + 3 strength = 6 * 3 Jump = 18 * 2 Step = 36 feet). Carry your opponent with you as you jump straight up.
At the top of your arc, release the grapple. And by "release the grapple", I mean hurl that poor guy back to the ground with a mighty heave.
Opponent falls and takes 3d6 damage, landing prone with no save.
You fall and take 3d6 - 5x Monk Level damage (assuming Slow Fall at level 4).
If you take no damage, use your remaining attack action from Extra Attack to re-grapple the prone target (attack #2)
If you take damage and land prone, use 5 feet of movement to stand from Athletic, and then regrapple the prone target.
BONUS: If your DM lets you, just land on your target; he should take 3d6 damage again from your fall.
I got this combo from the WotC Forums. The guide is called the Grappler's Manual, which also contains a section on wrestling combos toward the end. http://community.wizards.com/forum/player-help/threads/4142801