This post is sort of a non-answer, but here it is.
I’ve never heard the phrase or term “dollar-a-year” man before, but if I had run across it in the wild I would have assumed it referred to those public minded individuals who were donating their time for a truly nominal compensation. The law requires some exchange of value for a contract to be valid but does not enquire as to the relative value between the two sides of the exchange (except where fraud is suspected). Without some form of compensation, the individual’s contribution to the enterprise would be considered a gift, AFAIK. As a gift, it would probably have drastically different tax implications.
As Dave points out, the interpretations can be a little tricky. One can speculate all day long how much actual “donation” was involved when a person was working in a prominent or highly conspicuous position, doing actual work, all the while being seen as sacrificing his or her time. Conversely, there are those people who lend their names to a company but do nothing. And then there are the political appointees ... ugh.
The “dollar-a-year” aspect of this newer situation seems like a bastardization of an older, longer standing tradition. It also used to be fairly typical to “sell” a piece of property to a governmental agency, or between agencies, for the value of a dollar.