A: In membership meetings, the presiding officer may vote only when his vote would alter the outcome. One case, as you mention, is when a majority is required and the vote is a tie (thus failing). The officer could vote in the affirmative, breaking the tie and causing the adoption of the motion.
Another case is when a majority is required and the affirmative side has a one-vote lead (thus passing). The officer could vote in the negative, creating a tie and causing the failure of the motion.

(Exercise: If a two-thirds vote is required, and the vote is 10 in favor and 5 opposed, can the presiding officer vote? If so, which way?)

In boards of about twelve or fewer members, some rules are relaxed. If it is the rule or custom of these boards (as it is in most), the chairman may debate without leaving the chair, make motions, and vote regardless of whether his vote would effect the outcome.