If some unknown cause of red shift—such as light becoming “tired” from traveling long distances through empty space—is discovered, what effect would there be on cosmology?

The red shift type of phenomenon occurs due to the Doppler shift of the light wave due to the motion of the source relative to observer.

This would limit the ability to observe distant galaxies because light travelling through empty space would have a lower frequency, making it more difficult to observe as the galaxy got further away. It would also make calculating the relative velocity/acceleration from the expansion of a distant galaxy more difficult, because it relies on the fact that redshift is proportional to distance, which now has an effect to account for.

This effect, if strong enough, could imply that the universe is actually static (not expanding), because we would see a redshift proportional to distance from every point of view, which is one of the arguments used to support the "universe expanding" theory. Fortunately, no evidence of this effect has been found.