Phi (the Golden Ratio) is a convenient metric that has some fascinating coincidences, and that can be used for just about anything. I think it is important to keep in mind that Phi is a great guide, but it only just that: a guide. It is certainly worth playing around with if you are looking to learn a few new things. But, I think ultimately, subjectivity and rule-breaking is what leads to great design. Designing by the rules means you will have a design that looks like you designed by the rules. Have fun with Phi—I do all the time now—but don't rule your designs by it. The same could be said of any of the irrational mathematical constants.
I think it would have ultimately limited use in color selection, but the quality of those choices may be better than most, so it would be worth playing around with to see if there is some validity in this (and I would be very surprised if you didn't find someone who hasn't already done this work).
Looking at the typical color wheel, There are already several "algorithms" to choose from—complement, triad, tetrad, and the like—but most of those deal with straightforward numbers:.5, .333333 for the first two respectively. The remaining are more about relative location to each other over an exact proportion, and it is easy to come up with some wonky color choices. Even as useful as Phi can be, it can still lead you down some pretty horrible paths from a purely algorithmic perspective.
At the same time, tetrad, analogic, and accented analog I think offer the most fertile areas to play with Phi. The relative choices would diminish quickly mathematically, especially keeping within the 90˚ bounds the previously linked color wheel enforces. Again, maybe those overall quality of those choices are better or there is a better ratio of "signal to noise", and I think it would be interesting to see what the results are.