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)

Relevant cards

Note: All applicable cards may not be listed below.
XY9_49