Does Prayer Posture Matter?

On the one hand, no; prayer posture doesn't matter.  You can pray anywhere, anytime, and in any position.  Certainly, if you are praying while driving your car, by all means, keep your head up and your eyes open.  That's all well and good when you go about praying throughout your busy, hectic day.  I know of a chaplain in a youth detention center who learned very quickly to pray out loud with eyes wide open.
On the other hand, prayer posture matters a great deal.  This is especially true in your regularly scheduled times of prayer.  You know, when it comes time to go into your prayer closet, or even in a corporate worship service.  It all comes down our attitude as we approach the throne of God in prayer.  What the body does affects the soul.
Listen to what C. S. Lewis had to say about it through his infamously devilish character--Uncle Screwtape.
"At the very least, they (Christians) can be persuaded that the bodily position makes no difference to their prayers; for they constantly forget, what you (Wormwood) must always remember, that they are animals and that whatever their bodies do affects their souls."*
Bowing the head, kneeling, prostrating, hands lifted up, etc. are all actions of surrender and submission.  Submission is a wonderful idea when you're talking to God.  Through these actions, you're telling your body something, and in turn your bodily position will affect your mind as you pray.

*C. S. Lewis, The Screwtape Letters, The Complete C. S. Lewis Signature Classics (2002), 194-195.