I explained what an Arrow collar was, but didn't bother with "Coolidge dollar" because I figured there must have been a silver dollar in circulation with President Calvin Coolidge's puss on it. But my childhood friend Russell Handelman says he can find no evidence that any such dollar ever existed. Coolidge, he observes, had only died the year before the song was first performed,
and it would have been unlikely (what with the Democrats in control of the government at that point) for the Mint to have issued a commemorative dollar in time to be included in the song. Also, silver dollars were being phased out (the last ones were minted in 1935). So far as I can determine, via Google, the only silver dollars issued at that time carried the "Peace" dollar design instituted in 1921.
A numismatic mystery. But perhaps "Coolidge dollar" was meant in a more figurative and economic sense. A 1997 explication de texte for "You're the Top" published in Playbill by Louis Botto says that the "Coolidge dollar" refers to the strength of the United States dollar during the booming 1920s prior to the 1929 stock market crash, which occurred less than a year after Coolidge left office. That sounds plausible to me.
[Update, 8:30 p.m.: A reader points out that a silver Coolidge half-dollar, apparently minted while Coolidge was still alive, is for sale here at the price of $500. Is it possible Porter just got the denomination wrong?]