The opponent can tap their lands for mana in response to Mana Web’s ability triggering.

When a land an opponent controls is tapped for mana, this card checks which types of mana (white, blue, black, red, green, or colorless) that land could produce using any of its mana abilities, not merely the one the player just used. Then it checks which lands that player controls have abilities that could produce any of those types of mana. Then it taps those lands.

The lands that are tapped by Mana Web’s ability don’t produce mana as a result.