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For consistency and to make users recognize quickly clickable links/buttons, "the color of a button should depend on its state and its purpose"*.
In the case of the hamburguer button it does not behave as a link but it needs to call to action as links do.
Based on this I would think that it makes more sense to display it with the same color as the links, but in general I see this button in a different color than links, (more like in the text color) e.g.:
So what is the right path to follow?
In short, not unless you want it to.
Links are generally coloured so that they stand out from all the other non-linked text.
Buttons may have alternate colours to signify differences in function or state.
The hamburger menu is usually in an area of it's own and has a distinctive shape. It doesn't need to stand out from a lot of other hamburger-shaped items and doesn't really have any alternate sates or functionalities that might be presented together so there is no real need to colour it unless you really want to.
