The extra cost associated with Armor Ward is paid at the time that the equipment would be destroyed. This is usually during the Resolve Spell step. Hypothetically it could occur at other times though, such as the Damage and Effects step with a [mwcard=DNJ02]Corrosive Orchid[/mwcard].
Edit: To be perfectly clear in answering your original question, any incantation that currently exists and would destroy equipment would have to pay the extra cost during the Resolve Spell step. So, nullify would cancel it before the extra cost is paid. I just wanted to convey the information that this isn't because the step is set in stone for the extra cost, but rather that's simply the step in which those spells happen to destroy the equipment.
