The phrase I resemble that remark is a deliberate malapropism punning on I resent that remark—cf. also to cast nasturtiums.

The earliest occurrence that I have found is from The Kansas City Star (Kansas City, Missouri) of Sunday 29^th December 1940:

“MELODY RANCH.”
(Now playing at the Tower.)

The motion picture reviewers who saw “Melody Ranch” at the Hollywood premiere had the following opinions:
[…]
“Melody Ranch” is no ordinary Gene Autry ¹ western. It has many extra favors. Long-legged Tap-Dancer Ann Miller ² swings through a lively dance routine, croons a torchy ballad. Jimmy Durante ³ garbles his Bronxese with dialogue like, “I resemble dat remark! . . . . Dat’s libel . . . . It’s libel to make me mad!”
                                                                                       TIME MAGAZINE.

¹ Orvon Grover ‘Gene’ Autry (1907-1998), nicknamed The Singing Cowboy, was a U.S. singer, songwriter, actor, musician and rodeo performer.
² Ann Miller (Johnnie Lucille Collier – 1923-2004) was a U.S. dancer, singer and actress.
³ James Francis Durante (1893-1980) was a U.S. singer, musician and actor.

The second-earliest occurrence that I have found is from the nationally syndicated column It Happened Last Night, written by the U.S. journalist, columnist and author Harvey Earl Wilson (1907-1987), and published in July 1944, for example in the Evening Courier (Camden, New Jersey) of Friday the 14^th—Earl Wilson, who tells how he called on Lois Andrews ⁴ and unsuccessfully begged her to take George Jessel ⁵ back as a husband, specifies that I resemble that remark was already hackneyed:

“I want a little more security than a 20^th Century contract, a Racing Form and a hotel room,” she growled beautifully.
“I resemble that remark,” I said, stealing a line from several hundred masters of ceremonies. “Georgie has a beautiful home in Hollywood or Beverly Hills or somewhere.”

⁴ Lois Andrews (Lorraine Gourley – 1924-1968) was a U.S. actress.
⁵ George Jessel (1898-1981) was a U.S. actor.

The phrase then occurs in one of the Famous Fables cartoons, by E. E. Edgar, published in the Chicago Sunday Sun Times (Chicago, Illinois) of Sunday 3^rd October 1948:

OVERLOOKED: Like many another celebrity, tempermental [sic] actor Richard Mansfield ⁶ was no hero to his valet. One night, before going to the theater, he rang for his servant. When the latter did not respond promptly to the call, the actor became irritated.
“Where the devil is that blockhead of mine?” he cried.
“Just where it has always been,” said his valet, appearing in the doorway, “on your shoulders.”
“I RESEMBLE THAT REMARK!”

⁶ This perhaps refers to the English actor-manager Richard Mansfield (1857-1907).