Besides the difference between irse a as expression of movement and ir a as a periphrastic future tense, there's another nuance that I discussed a bit in this question about reír as a pronominal verb.
I would say in fact that Me voy a dormir doesn't sound to me as if the movement (me towards my bed, presumably) is emphasized. Indeed you might say Me voy a dormir even if you're already sitting on your bed. Or you might say Me voy a casa a dormir and there you have dormir as intent instead of movement (a dormir here means the same as para dormir).
What I'm suggesting is that the pronominal element in irse a dormir is not really so much part of irse as a verb of movement, but the same element that appears in dormirse.
Vámonos que me duermo. ("Let's go, [because] I'm getting asleep.")
Me dormí en el trabajo. ("I fell asleep at work.")
Esta noche duermo solo. ("Tonight I sleep alone.")
Dormí como un bebé. ("I slept like a baby.")
There are quite a few verbs where the pronominal form is used for mediopassive meanings (actions that affect the speaker) while the simple form has a more general, neutral meaning. (Dormir just means "to sleep", i. e. to sleep in a certain way, in a given place, for a certain amount of time, etc.; dormirse means "to fall asleep", "to get sleepy", "to get [oneself] a given quality of sleep").
In this same sense, Voy a dormir just asserts that you're going (somewhere, sometime) to sleep, while Me voy a dormir is rather like saying you're going away from other people into your own, personal, sleepy bubble.