The rule could be anything, as funny or as serious as you want. The universe will progress in a similar way that it has up until this point, unless your changed rule prevented it from doing so.
Some examples might be:
- The invention of currency is not allowed.
- Iron is slightly less stable.
- The Ancient Greeks are able to cultivate Silphium, which does not go extinct now.
Since the speed is infinite, the wavelength is always zero, no matter the frequency. But a wavelength of zero at infinite speed means an infinite number of waves hit the eye in any constant amount of time. Hence, and infinite amount of energy hits the eye at practically the same moment. But assuming your eye (and all of the universe) does not incinerate instantly, you will not be able to see the difference between red and blue, since they are perceived to have the same wavelength (zero). So yeah, technically you no longer have blue/redshifting.
OK, I edited my comment. I hope you don’t find any more scientific proof against that too!