Also known as a conspiracy board, crazy wall, murder map, conspiracy wall, link chart, investigation board, etc.
The earliest appearance I can think of is in the first episode of Tinker Tailor Soldier Spy (1979).
You can see it nine minutes into the episode:
The Vault of Culture has over 800 posts tagged "Narrative String Theory" which describe works that loosely fit, though some like The Omen (1976) are more collages than what is sought in this question, since there is no indication of connections between the items on the walls. There are earlier works, including Duchamps' art installation Sixteen Miles of String (1942), Philip K. Dick's novel The Broken Bubble (1956) and even the 1375 Catalan Atlas, but none of them are movies or TV shows (or even plays).
I found a Slate article about an FBI recruitment ad that used a string board. The article discusses string boards/pinboards (even linking the String Theory trope) and their origin in popular culture, with this paragraph notable:
Certainly, it's a "visually striking attribute usable for anything from 'genius mind at work' to 'very complicated situation' to 'crazy obsessed person,'" as Anne Ganzert put it in her 2020 book, Serial Pinboarding in Contemporary Television. But no one can quite pinpoint when this trope emerged. The earliest version of it seems to have come from the 1979 BBC production of Tinker Tailor Soldier Spy, which had an understated series of red lines connecting photos to a calendar in one episode. It's unclear how it might have taken off from there.
It may be unclear how it took off, but scrolling through the 800+ entries in the Vault of Culture list, I note relatively few appearances between 1979 and 2000, and a dramatic increase in its appearance in the years since A Beautiful Mind (2001) came out.