The origin of the "three dollar bill" being irredeemable might be traced to the failure of the Kirtland Anti-Banking Society in the 1830's and the three dollar notes it printed. The principals of this anti-bank bank were arrested, lost in court and fled the state of Illinois, incurring perhaps as much as $150,000 (in 1830's US dollars) in losses to the investors.
A picture of the "three dollar bill" can be found (among other places) on page 35 of
John L. Smith's 2006 Rigdon book.
There were other, earlier, "three dollar bills" printed by banks in the US; however, to my knowledge, none of those institutions had as much scandal (irredeemability) associated with them as those printed by the Kirtland Anti-Banking Society in the 1830's. A picture of a three dollar bill from the 1700's as well as the historical context of such bank-issued notes can be found at
http://www.common-place.org/vol-04/no-04/mihm/2.shtml