Finley Peter Dunne was a Chicago newspaperman who wrote a editorial
column in the persona of "Mr. Dooley." I believe it was for the
Chicago Tribune.  My question is which day of the week these columns
appeared?

The "Mr. Dooley" columns appeared on Saturdays.

"It was at the Chicago Evening Post where he created the character of
Martin J. Dooley, who was modeled after a real-life Chicago bartender.
Mr. Dooley was known for his quick wit, common sense, and strong
opinions delivered in a broad Irish dialect.

Dunne's weekly columns became a Saturday evening tradition in Chicago.
Between 1893 and 1900, he wrote about 300 Dooley pieces."

Chicago Tribute: Finley Peter Dunne
http://www.chicagotribute.org/Markers/Dunne.htm

To check this, I looked up the publication dates of several "Mr.
Dooley" columns. Using an online 10,000-year calendar, I verified that
each column appeared on a Saturday.

My Google search strategy:

Google Web Search: "finley peter dunne" "mr dooley" weekly
://www.google.com/search?hl=en&q=%22finley+peter+dunne%22+%22mr+dooley%22+weekly

I hope this is helpful. If anything is unclear or incomplete, please
request clarification; I'll be glad to offer further assistance before
you rate my answer.

Best regards,
pinkfreud