"I request you to change the color of the book cover from the blue to the white."
The entire sentence, as written, sounds wrong; however, using an article with colours in similar contexts can be correct.
You can abbreviate statements about things if you have previously stated certain details already. For example, if a restaurant menu had beef bourguignon on, and it was the only beef dish, you could just say "I'll have the beef", without needing to state its full name.
Similarly, if you had two pens - one red, one blue - and you had already established that you were talking about the pens, you could just refer to them by the colours, for example:
Q. Which pen do you want - the red, or the blue?
If, in your example, the 'covers' were interchangeable objects (ie you have two replaceable covers for the book, one white, one blue) then there is no problem referring to them by their colours. This would be correctly written as:
"I request you to change
the color of the book cover from the blue to the white."
Your example doesn't seem right because it also uses an article in "the book cover", suggesting that there is only one. You could change the colour of something by applying paint, in which case your example would be wrong and the articles should be removed because the colours are not objects requiring an article.