I am not sure how to interpret this situation. I suppose that a player could give a note to the Nekro player and demand that he abstain from the vote, but that doesn't make much sense either. Thoughts?

Nothing says they can't. Though I can't see why someone would normally give them one, they are giving the Necro Virus a powerful binding agreement in exchange for nothing or an essentially non-binding agreement. Now if the Necro Virus got hit with Cultural Crisis the Note giver would gain the usual benefits, but that is sorta rare.

If someone expect vote to be tied and necro players is holding speaker token - then guess what. Nekro player is picking the outcome of the vote.

This might be one of the reasons why someone would give them one.

But breaking the tie is not "voting." Otherwise a Necro Virus Speaker would be unable to break a tie leading to a stalemate. Promissory Notes only set what a player player votes, breaking a tie as the Speaker is outside of that.