Hi,
Yes, the word can be used in this way to suggest a problem, an annoyance at injustice or perhaps more specifically, irritation.
So for example, someone might be talking about a colleague and say something like, "the way they always take credit for other people's work really rubs me up the wrong way." Another example being use of the word grate. Although technically this could refer to the action of grating food, or metal covering (like a floor grate) it can also be used to suggest irritation in the same way as rub, in that somebody you know may have an annoying laugh which you might say "is beginning to grate on me."