If I play Enchanted Evening and turn everything into an enchantment, and then play Grand Abolisher to disallow them activating abilities during my turn, can my opponents tap their lands during my turn (say, to make a payment for something)?
No they cannot.
From the rulings on Enchanted Evening:
Which Grand Abolisher clearly interacts with:
I think it's clear that lands have an innate implied ability, even if not printed of "Tap: Add 1 mana of the appropriate colour to your mana pool."
So all we need to know is that this ability is treated like any other:
The exceptions listed don't apply to this case because Grand Abolisher is not countering the mana ability, it is preventing it from happening. Just because it's a mana ability doesn't make it special in this case.