As with all novels from James's so-called "major phase", the symbolism of The Wings of the Dove is complex and multi-layered. F. O. Mathiessen, one of the most important American critics of the first half of the twentieth century, has written interesting pages on the novel's symbolism, which I summarise here and that you can read in full following the second link below.
The image of the dove evokes images of purity and innocence. This is certainly one ways in which the narrator sets up an opposition between Milly Theale and the more manipulative characters such as Kate, her aunt Maud (who is compared to an eagle with "gilded claws") and her lover Merton. It is Merton who first sees Milly as a "Christian maiden, in the arena, mildly, caressingly martyred" by "domestic animals". Yet, the narrator immediately complicates the symbolism quoting Milly's own self-description of having used "the wisdown of the serpent" to find the doctor that can best address her illness. In a later scene in Venice, the image of the dove itself is shown as not only denoting innocence, but also ability to conceal feelings and the truth. Finally, the whiteness of the dove is also linked to Milly's pearls thus symbolizing the power that her wealth gives her. The wings are also used as a symbol of protection.