There's a card called "Skill Drain" which negates the effects of monsters, just like Axe of Fools. And that's how it was ruled. The effect of losing half ATK/DEF is negated, but it's still destroyed at the End phase.
Effects can still be activated, so even with Axe equipped you can still activate the Emperor's effect. You will still need to banish one card (because it's a cost, which is not negated), but you will NOT be able destroy one card (because it's an effect). In short, don't use his ability, or you'll just end up throw ling one card away (not to graveyard, but banished).