I've never messed much with tasks, but regular calendar events have issues in Tbird/lightning: If you create a calendar entry with Lightning, you can move, delete, change it with Lightning and also view it on Zimbra. But the moment you actually modify it from the Zimbra web client, you lose control over it in Lightning. You can still see it, but any deletes or moves don't take.
The issue was discussed last year on
this thread. I'd swear there was a bug filed on this, but I can't find it. Part of the issue is that we don't know to what extent the problem is Zimbra or Lightning.
I had kinda let the whole thing ride because of upcoming CalDav implementation in 5.x, which
according to the wiki is now true. I've been too busy to upgrade yet at my shop so I have not had time to check it out, but this may fix the problem. . .at least I sincerely hope so!
Cheers,
Dan