Altaria has a printed weakness to Lightning, but it also has the "Clear Humming" Ability which creates an effect that gives it no weakness. What happens if it is attacked by Zebstrika with the "Zap Zone" Ability which makes its attack's damage ignore effects on the defending Pokemon? When Zebstrika attacks Altaria, is the damage altered by weakness or not?
Zap Zone refers to "any effects on your opponent's Active Pokemon", but the term "any effects" does not include effects that alter Weakness or Resistance. So in this case Zebstrika's attack will NOT be increased for Weakness, due to Clear Humming.
Source: TPCi Rules Team (2016-03-24), TPCi Rules Changes Announcement (2017-05-03), TPCi Rules Changes Announcement (2017-05-18)
If my Zebstrika with the "Zap Zone" Ability attacks a Pokemon with {L} Weakness, does the Weakness get ignored because of "Zap Zone"?
No, Weakness & Resistance are game mechanics, they are not effects that would be ignored because of Zap Zone.
Source: BREAKpoint FAQ (2016-01-23), TPCi Rules Team (2016-02-11)