meaning - What is the origin and sense of the phrase “put up or shut up”?

In researching the recent EL&U question Origins and meaning of "Put your money where your mouth is", I repeatedly came across the seemingly related but older phrase “put up or shut up.” Where and when did this older phrase arise, and what did it originally mean?

I will submit the information that I’ve uncovered so far as an answer below.

Answer

The phrase comes from boxing, when one fighter would challenge another and require him to put up a stake for a match, or stop his fighting words.

1858

The OED's earliest quotation is:

1858 Marysville (Ohio) Tribune (Electronic text) 21 July, Now, if he means business, let him put up, or shut up, for this is the last communication that will come from me in regard to this fellow.

Here it is printed in another newspaper with more context:

The Brutality of the Nineteenth Century—Somewhat after the manner of duelling correspondence, two pugilists, John Morrissey and John Heenan, the latter known as the "Benicia Hoy," have been negotiating for a fight. — Their letters are published in Porter's Spirit of the Times, where it is openly announced that they are to fight in Canada on the 20th of October, for $2,500 a side, the exact place to be named by the editor of "a city paper." Morrissey, who gave the challenge, closes his letter as follows:

"There has always been some objections, however, to every proposition I have made ; some little quibble that this man Heenan has raised, with how much courage and manliness I leave your readers to judge. The above proposition is certainly a fair one, and no man can object to it.— Now, if he means business, let him put up, or shut up, for this is the last communication that will come from me in regard to this fellow."

And the other ruffian, in his reply, says that he has resigned a place under government in order to accept the challenge, and concludes :

"Mr. Morrissey's slang terms are beneath my notice, as I prefer to conduct the matter in a gentlemanly and straightforward manner."

Cambridge Chronicle, Volume XIII, Number 32, 7 August 1858

1865

The earliest I found in the Chronicling America archive from 1865 makes the meaning clear:

If Reilley concluded to accept my proposition, he will find two hundred dollars deposited at the Fountain Restaurant, with John Cornell and Billy Nuttall, California Exchange, to be put up as a forfeit. So put up or shut up.

The Montana Post, July 15, 1865 (Virginia City, Montana Territory [i.e. Mont.])

1867

The next I found is also a boxing challenge in The Montana Post, November 09, 1867. After that, it's used referring to a bet: put up your money to take on the bet or shut up.