Heat illnesses are about heat, not light, and while the two are not unrelated, the hue of your clothing would be a very minor factor— red would not provide better or worse protection than green or blue or any other part of the visible spectrum.
The shade may have some impact: since darker clothing absorbs more energy than lighter clothing, it warms up and stays warm in sunlight. Various guides will advise wearing light colored clothing— better a light red than a dark blue, but better a light blue than a dark red.
But heat stroke can be a danger even for a well-hydrated athlete even on an overcast day. Breathability and fit are more important for choosing heat-appropriate clothing, and clothing is only one consideration among many.
(answer for previous question about sunburn)
Perhaps one would wear red clothing to hide a sunburn, or make it less noticeable. For prevention, however, there is no scientific basis to prefer any color. Sunburns are caused by ultraviolet light, whose waves are outside the spectrum of human vision. The hue of whatever garment you are wearing has no bearing at all; you can even purchase black UPF-rated clothing.
Rather, a garment's thickness, the tightness of its fabric weave, and the UV-reflective qualities of its dyes are more important.