Here's what I remember: The protagonist has a friend who is an artist. A man (art critic? art historian? maybe even sociologist?) comes from the future all gushing about how this friend's art is the basis of their whole future society. While the critic is busy inspecting the art, the artist absconds into the future with his time machine. The critic then has to assume his identity and paint all of the art he came back to the past to examine.
This is ‘The Discovery of Morniel Mathaway’ by William Tenn, first published in Galaxy Science Fiction, October 1955. The original publication of the story is available via the Internet Archive, and it confirms the plot points you remembered:
“A man (art critic? art historian? maybe even sociologist?) comes from the future all gushing about how this friend’s art is the basis of their whole future society.”
A series of high musical notes sounded, one after the other, rapidly. And then, in the center of the room, about two feet above the floor this time, the purple lines reappeared still hazy, still transparent and still with the outline of a man inside. […]
“Morniel Mathaway,” the man from the box said, “my name is Glescu. I bring you greetings from 2487 A.D. […] In my own period, I may say without much fear of contradiction, I am the greatest living authority on the life and works of Morniel Mathaway. My special field is you.”
“While the critic is busy inspecting the art, the artist absconds into the future with his time machine.”
The book wasn’t there; the bed was empty. And two other things weren’t there—the time machine and Morniel Mathaway.
“He left in it!” Mr. Glescu gasped. “He stranded me here! He must have figured out that getting inside and closing the door made it return!”
“The critic then has to assume his identity and paint all of the art he came back to the past to examine.”
And then I got the idea. “No, it needn’t be. Tell you what. Morniel has a social security card—he had a job a couple of years ago. And he keeps his birth certificate in that bureau drawer along with other personal papers. Why don’t you just assume his identity? He’ll never show you up as an imposter!”
The story was popularised as an exemplary paradox of causality by philosopher Michael Dummett, who summarized it without mentioning the title or author in a 1986 book chapter. Probably Dummett could not remember these details about a story he had read a long time previously, but however it happend Tenn was deprived of the credit for the ideas about causal paradoxes.
If there are causal chains running in the reverse as well as in the usual direction, there is a possibility of causal loops. They often occur in science fiction: one such story that amused me concerned a fifth-rate but conceited artist. One day he is visited by an art critic from a century ahead,† who explains that he has been selected for time travel so that he could interview the artist, who is regarded, in the critic’s time, as by far the greatest artist of the twentieth century.
Michael Dummett (1986). ‘Causal Loops’. In Raymond Flood and Michael Lockwood, eds. The Nature of Time, p. 155. Oxford: Basil Blackwell.
† Sic: in light of the “2487 A.D.” date in the story, this should be “five centuries”.
Several other philosophers have copied or summarized this description from Dummett, perpetuating the lack of proper credit for the story. These include Michael Lockwood (2005), The Labyrinth of Time; Friedel Weinert (2013), The March of Time; and Storrs McCall (2014), The Consistency of Arithmetic.