I have 2 {Ancient Tomb} and 1 {Platinum Emperion} in the battlefield, when I tap both of my ancient tombs, my opponent response with a {Swords to Plowshares} on my emperion. Do I take 4 damage from tomb and gain 8 life from my emperion? Or tapping my lands do not use the stack; he has no priority to cast instant during my main phase?
No he cannot. The damage dealing part of Ancient Tomb is tied into a mana ability. It resolves immediately as you place the mana into your pool. Your opponent cannot respond to mana abilities. As a result, your life doesn't change.
Ancient Tomb has the damage as part of the mana ability, so it can't be responded to. If it were {City of Brass}, however, they could respond to the damage ability on the stack, as it is a separate triggered ability.