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Let's say that I cast Goblin Grenade, sacrificing a goblin to pay it's additional cost. Then, before it resolves I cast Increasing Vengeance, which allows me to copy the Goblin Grenade. Since I've already paid the additional cost of Goblin Grenade, do I have to pay it again?

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No, you do not need to pay additional costs for a copied spell. A similar example of Fling exists in the comprehensive rules, that is close enough to Goblin Grenade for illustration purposes.

706.10. To copy a spell or activated ability means to put a copy of it onto the stack; a copy of a spell isn’t cast and a copy of an activated ability isn’t activated. A copy of a spell or ability copies both the characteristics of the spell or ability and all decisions made for it, including modes, targets, the value of X, and additional or alternative costs. (See rule 601, “Casting Spells.”)

Example: Fling is an instant that reads, “As an additional cost to cast Fling, sacrifice a creature” and “Fling deals damage equal to the sacrificed creature’s power to target creature or player.” When determining how much damage a copy of Fling deals, it checks the power of the creature sacrificed to pay for the original Fling.

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You don't need to sacrifice another Goblin or pay any other costs.

Additional costs are paid as part of casting a spell. When you copy a spell with Increasing Vengeance, the copy just pops into existence straight on the stack. Technically speaking, it's never "cast" at all.

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  • If you put a copy of a spell on the stack, you can't pay regular, alternative, or additional costs. It's not a choice. This is relevant when you don't copy an existing spell but put it on the stack by other means, such as Isochron Scepter. For example, a Fling put on the stack through an Isochron Scepter Imprint cannot deal damage because you can't sacrifice a creature to it. – Hackworth Jul 13 '12 at 14:57
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    @Hackworth Where you'd going is correct, but Isochron Scepter is an incorrect example. The scepter doesn't copy a spell, it casts a copy of a card, Fling works fine on it. – Affe Jul 13 '12 at 19:05
  • @Hackworth The rulings on isochron scepter very specifically state you can pay additional costs. Of the 145 cards that reference casting spells or copies of spells without some kind of payment, every one of them specifies that it is done without paying the mana cost - any additional costs must be paid. – Andrew Feb 19 at 15:00
  • The same is true for the 129 cards with the keywords Cascade, Cipher, Rebound, Ripple, and Suspend which all allow casts without paying, and specify mana costs in their rules. There is no current effect in magic that allows the casting of spells completely without the payment of any costs, simply most cards only have a mana cost to prevent and most people start to think of that as the whole cost. – Andrew Feb 19 at 15:08

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