See
The Universe at Midnight by Ken Croswell, pages 75-76, on the tired-light theory:
The tired-light theory is not new. It was first proposed by maverick scientist Fritz Zwicky in 1929, a few months after Hubble discovered the distance-redshift relation. But two observations rule it out. First, astronomers see that exploding stars in distant galaxies brighten and fade more slowly than those nearby. This time dilation arises from the expansion of space. To see how, imagine that a star in a far-off galaxy emits one pulse of light toward Earth on January 1 and a second pulse on February 1. Initially, the two pulses are separated by a distance of one light-month. As they travel toward Earth, though, the space between them expands, perhaps doubling; so astronomers receive them two months apart. In the tired-light theory, this should not happen–the pulses of light weaken but do not separate. In fact, astronomers do observe that distant supernovae wax and wane more slowly than nearby ones, agreeing with the idea that space expands and contradicting the tired-light theory.
Second, the tired-light theory disagrees with the observed spectrum of the cosmic microwave background, the big bang's afterglow. This has a specific shape which physicists call a blackbody: it is most intense at one particular wavelength, falls off slowly at longer wavelengths, but rapidly at shorter wavelengths. The universe's expansion degrades the cosmic microwave background's spectrum, stretching it to longer wavelengths, but in a way that preserves the blackbody shape. In contrast, the tired-light theory predicts that as the light composing the cosmic microwave background loses energy, the spectrum ceases to remain a blackbody, contrary to observations.
--from
The Universe at Midnight by Ken Croswell. Link:
The Universe at Midnight