I don’t have a confirmed case of Linux running without a traditional CPU, but I wouldn’t bet $20.00 against it existing
There’s plenty of devices out there that do some kind of computation and don’t have anything we would recognize at a traditional CPU. Such devices are becoming rare, because CPUs are so dang cheap, now.
Most of those run binary compiled from custom C code or even directly from Assembly.
But, if someone was going to run an alternative program on one of those devices, there’s a very good chance they would install Linux (heavily customized) on it first.
Source: I’ve completed parts of Linux from Scratch. It was eye opening to realize the places that a true Linux expert could get it to run. Every single part of Linux is truly optional, to someone who knows that they’re doing.