Is it permissible for a person working in charitable work to say that he does not take a wage—while he does—in order to ward off annoyances and embarrassments, knowing that the work is under his supervision?

If the questioner is a supervisor and director of a charitable work who receives donations, then he is not permitted to take from them unless the donors are aware. However, if he is appointed by the administration to supervise in exchange for a known wage (salary), then there is no harm in that. Reporting something contrary to reality is a forbidden lie, and there is no necessity in the questioner's situation that permits lying. If there is a need, then circumlocutions (ma'areedh) suffice; these involve using a word that has more than one meaning so that the listener understands something other than what the speaker intends.