It isn't thought to be serious just yet -- manager Bobby Valentine described the discomfort as "little" and focused more on stiffness in said elbow than any kind of explicit pain. Until we know more, we don't know how this will effect his chances at making the Opening Day roster.
Miller is out of options along with a few other pitchers vying for jobs -- Matt Albers, Felix Doubront, and Michael Bowden -- and while he can be listed as one of the Rotationettes, he was more likely looking for a gig in the bullpen, since it's the more direct route to the majors, and would keep the Red Sox from needing to put him through waivers to send him to the minors. Miller's inconsistent mechanics and command make him a better fit for a pen role, and the Red Sox will need arms capable of going a few innings of relief at a time if Daniel Bard (or both Bard and Alfredo Aceves) make the rotation, anyway.
We're not quite at the moment where there's more pressure on the likes of Felix Doubront and Daniel Bard to step up this spring and secure rotation gigs, given Miller was on the lower rungs of that particular ladder. But hopefully that's the end of even these minor setbacks for a while.