The Dark Chamber (Leonard Cline, 1927) contains the inverse of this concept, where the main character attempts to recall the memories of his ancestors, one generation at a time (i.e., passing memories forward instead of passing them back). The cumulative memories eventually outweigh his own, taking over his personality (he essentially "becomes" his ancestors, or rather, they invade his mind/body). When he passes beyond the species threshold of humanity, he essentially devolves into that of a proto-human, ultimately destroying his household and family. This "hereditary memory" concept could be construed as a form of forward time-travel of his ancestors.
I'm going to posit that this is the original concept (forward time travel via memory) that would lead subsequent authors to the idea of this process running in the opposite direction. Given this other answer, that puts the origin of the idea between 1927 and 1947.
A partial review from Time Magazine, September 5, 1927 (c.f. here for the full review):
Fall of the House of Pride
Author Leonard Cline has been weaving spells in the vicinity of
Englewood, N. J. He has regrown the virgin forest there and placed in
its tangled heart a mansion full of dark madness.
In Mordance Hall lives Richard Pride, whose madness is living life
over again, living it beside himself with audacious backward
excursions into the lives of people he has known. This he does by the
aid of drugs and a corps of skilled, secret investigators. In his
sepulchral study, entombed by the inky transcripts of his assistants, he traces bizarre designs through the dead mold of past existence.
An unusual end to the book review lends some context to his mindset when he wrote the manuscript:
The Author is in prison. One dawn last summer, police found him in a
daze in front of his home at Mansfield, Conn., with a discharged
shotgun in his hands. Within lay one Wilfred Peter Irwin, shot in the
back, dying. Both men had been drinking for days. Before the guest
died he swore his host was innocent, the shooting an accident. But
Leonard Cline must stand trial for murder. Until the plot of that true
story is unraveled next month before a grand jury, one of the most
promising careers in U.S. literature is in abeyance. Factitious folk
have tried, futilely, to draw conclusions from the identical first
names of Mr. Cline's unfortunate guest and one of his minor
Prior to turning novelist, Leonard Cline wrote on art, music and books
for newspapers in Detroit, St. Louis, Baltimore, Manhattan. He was
born in Bay City, Mich., 34 years ago .
The book was highly influential on contemporary authors due to accolades from none other than H.P. Lovecraft. From Wikipedia:
[H.P.] Lovecraft passed his copy of The Dark Chamber to several of his fellow
writers, including Clark Ashton Smith, Donald Wandrei, Frank Belknap
Long, and Henry S. Whitehead. All of these writers later wrote fiction
about hereditary memory that were influenced by the novel.
It was also listed by horror critic R.S. Hadji in his list of
"unjustly neglected" horror fiction.
The original source of the above quote (Weird Tales - October 9, 2006) goes on to state that Cline was the first to come up with the concept:
All of these wrote stories dealing with ancestral memory that owe
their existence in large part to Cline, but Cline’s treatment of the
theme, besides being the first, is also the most powerful. Cline used
the Gothic trappings as a cultural shorthand that allowed him to
communicate ideas that were cutting-edge during the 1920s.
He/she goes on to explain why it is so difficult to find references to his works:
He is an excellent example of how mainstream writers prior to the
1920s could utilize themes and subject matter that would be thrust
into a literary ghetto in a few years with the rise of critics such as
Irving Babbitt and Edmund Wilson.
For another (2009) contemporary review of the book, see here.