It's queer as a three-dollar bill. There used to really be two-dollar bills. The idea originated back when people were more often confronted with fake bills. They are not counterfeit, because they do not actually match U.S. currency; there never has been an authentic three-dollar bill. But if you saw a three-dollar bill, you knew it was "queer," that is, fake. (This before the word "queer" had become exclusive to homosexuals. And long before "gay" got assigned to them, too, of course.)
My grandfather had a three-dollar bill, which was actually a promotional gimmick -- an advertisement. He said some people used to be so naive (or perhaps illiterate) they would think it was real money and try to pass it. The picture on it was of a chiropractor! (I wonder if he ever thought it would be used as real money!) Apparently the advertisement was that he would do an adjustment for $3.00, which tells you how old the thing had to be.
It's like the thing with wooden nickels: they were for specific towns, or even just Main Street of the town, to use as a sort of "coupon." People would give away the wooden nickels, and you could redeem them when you shopped. So "don't take any wooden nickels" should probably read "don't take JUST ANY wooden nickels" -- only the ones for your Chamber of Commerce, or whoever issued it.