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The following is meant to give a general explanation of the phenomenon of the orange colouration that is found in most goldfish and plenty of other species fish (as well as many reptiles and fish). It starts by giving a general explanation behind the origin of this colouration and proceed to discuss the various factors by which it can potentially be affected.
The orange colour in goldfish is mainly/entirely the result of the deposition of carotenoids in or right beneath the scales. Carotenoids are widely used in plants to protect the inside of the leaves from strong sunlight (towards higher frequencies, such as blue light and UV light, because these can cause the most harm). Animals also use them, either as 'anti-sun' pigmentation or as a colour display, to signal e.g. quality towards members of the opposite sex or against competitors of the same sex (many bird species are famous for this), and they also have some function in antioxidant activity (though it's still a bit unclear to what extend these are important). Importantly, most (if not all) animals cannot produce carotenoids on their own; they have to eat plant to acquire them. The deposition of the acquired carotenoids in feathers, skin or scales depends on quite a lot of factors: amount of available carotenoids, the amount of light exposure, and of course genetic factors pertaining to allocation of carotenoids to the skin/feathers/scales.
In birds, I know that once carotenoids are used as pigmentation in the feathers, the birds cannot remove them actively. In goldfish I'm not sure whether or not they can remove the carotenoids after deposition (I do not know of any research that has been done with regard to this), but there does seem to be some effect of both nutrition (e.g. how rich their food is with regard to carotenoids) and sunlight exposure, though from what I have surmised from other forum members, it's rather hard to use this in any definite way to actively change the colour of your goldfish (apart that sunlight seems to 'deepen' the colours that are already present).
In theory, a carotenoid-poor diet should decrease the amount of carotenoids in the scales, since (if they're not actively removed) they still need to be replaced alongside general scale replacement/growth. Little-or-no exposure to sunlight might decrease the colour, either if the carotenoids are actively removed (and allocated to other used when not needed as pigmentation) or if light is needed to allocate new carotenoids to the skin (like humans mostly only tan when exposed to plenty of sun). This however assumes that sunlight (alongside nutrition) is the only factor influencing carotenoid allocation; it might however be possible that mate attraction plays a (large) part in this ("look at me, I'm so very orange, so you know I'm a high quality male/female") or that the tendency to allocate more vs. less carotenoids to the scales is partly or mostly genetically determined (or in combination, that the mate-attraction-based allocation is genetically determined).
In the case that carotenoid deposition is (entirely) genetically determined it might be possible to use a breeding program to come up with 'genetically white' goldfish, that stay white no matter what diet or light they are exposed to. Because of the intricacies of genetics, it may be that this can be achieved relatively quickly (in only a couple of generations, if you wisely start with goldfish that are already on the light side) or it may be (at much more likely) that you would need a very stringent breeding program to achieve a breed of goldfish that is relatively unaffected by diet of light exposure, in the sense that if you stay clear of very carotenoid-rich food and high light exposure, you usually get entirely white goldfish. I should also be noted here, that there is a different between selection for a certain trait (e.g. lack of carotenoid pigmentation) and no selection against that trait: even if carotenoid colouration is no longer required (e.g. no bright light and no role mate attraction), that trait may still remain in the population, either because it is neither positive nor negative with regard to survival or reproduction, or because the mechanism behind carotenoid allocation in intertwined with other mechanisms that are positively selected for (in which case carotenoid colouration is just 'long for the ride'). Only traits that are (practically) neutral with regard to survival and reproduction may nevertheless change over time/generations (a process called genetic drift)
To summarise, carotenoid based pigmentation is a rather complex phenomenon, that can potentially be influenced by a lot of factors, either environmental, genetic or evolutionary. Although it's as such a very interesting topic, it's therefore also relatively difficult to apply to your own goldfish, although general rules can be applies to maximize the chances of your goldfish's colouration becoming what you want it to be.
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