No the ability stays until the match is over, if the ability went away when the leader died, most of the people would quickly try and take down the leader such as wedge or vader
Leader (and ally!) abilities are always on, even if the character is dead. Some aspects of the ability (like Vader's DoT auto-refresh) might be character dependent. For the most part, anything that's dependent upon the character being alive will be put into a Unique Ability, like Wedge's Red Leader ability.
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I 7* FOTP without using any energy refreshes, get on my level
https://swgoh.gg/characters/darth-vader/
Vader's Zeta is indeed only active while he is alive...all the more reason to take him out first.