Of course it will, and I think it will for all players, just by different amounts.
I read Brian Towers's answer. I had noticed the same thing in videos by Danya Naroditsky: When calculations get intense, it covers his eyes or looks away. That amazes me. And I've heard super GMs calculate and they hold the position in their head amazingly well (some better than others, of course. Fisher, Anand, and Kasparov are especially renowned for this).
But even for them, I am sure having an analysis board will help. Why am I sure? Because all these players use analysis boards when they are analyzing at home. It's true that good players can read chess books without a board much better then lower rated people, but even they do better when they have a board.
Also, I have heard some commentary about correspondence chess between top players that indicates that they get much farther along in variations then they do OTB.