A politician can’t suffer from “foot-in-mouth disease” (a gaffe) if he or she doesn’t talk (have an “open mouth”). The proverb “a closed mouth gathers no feet” is taken from the saying “to put one’s foot in one’s mouth” structured like the saying “a rolling stone gathers no moss.”
 
Bob Cooke, the sports editor of the New York (NY) Herald Tribune, is given credit for coining “a closed mouth gathers no feet” in 1956. The new proverb is also written as “a closed mouth gathers no foot.” One variant is “a closed mouth gathers no flies.”
 
   
Wiktionary: a closed mouth gathers no feet
Etymology
From put one’s foot in one’s mouth, adopting the structure of a rolling stone gathers no moss.
Proverba
a closed mouth gathers no feet
1. One who does not speak can be certain he won’t say anything embarrassing.
 
Wikipedia: A rolling stone gathers no moss
Taken literally, a rolling stone gathers no moss accurately describes that moss will not grow on a piece of rock which is moving by revolving or turning over and over. However, the phrase is most commonly used as a proverb.
 
Proverb
A rolling stone gathers no moss is an old proverb, credited to Publius Syrus, who in his Sententiae states, People who are always moving, with no roots in one place, avoid responsibilities and cares. As such, the proverb is often interpreted as referring to figurative nomads who avoid taking on responsibilities or cultivating or advancing their own knowledge, experience, or culture. Another interpretation equates “moss” to “stagnation”; as such the proverb can also refer to those who keep moving as never lacking for fresh ideas or creativity.
 
In English
The conventional English translation appeared in John Heywood’s collection of Proverbs in 1546.
 
Zazzle.com
A closed mouth gathers no foot.
tee shirt
 
New York (NY) Times
Bob Cooke, Sports Editor, 75
AP
Published: May 09, 1989
RIVERHEAD, L.I., May 8 — Bob Cooke, the sports editor of The New York Herald Tribune from 1945-65, died Sunday. He was 75 years old.
 
Mr. Cooke left The Herald Tribune to become managing editor of Golf magazine. He later worked as public-relations director for Schaefer Brewery, then with Harry M. Stevens, the ball park food-service company.
       
8 March 1956, Oakland (CA) Tribune, “The Jester,” pg. 59, col. 5:
AND IN CONCLUSION
A closed mouth gathers no feet.—New York Herald Tribune.
 
Google Books
The Reader’s Digest
Volume 68
1956
Pg. 78:
Bob Cooke: A closed mouth gathers no feet. 
—New York Herald Tribune
   
13 April 1958, Aberdeen (SD) American-News, “Jottings from the Dakotas” by Max Cooper, pg. 23, col. 1:
The article on a “Bistro for the Brain Bunch” contained additional information. One imbiber, who probably learned the hard way, wrote that, “A Closed Mouth Gathers No Feet.”
 
28 February 1959, Christian Science Monitor, “Extra $1,000 Probably Riding on Tomorrow’s Celtics-Hawks TV Game” by Phil Elderkin, pg. 13:
Possibly on the theory that a closed mouth gathers no feet, Coach Red Auerbach has been understandably reluctant to discuss the formula which has produced ...