The broader question is how can we allow the name of hashem to appear on a computer screen at all when we know the screen will eventually be turned off or the text moved. Whether you daven from the screen or not the issue is the same.
There are a number of sources that deal with this, and the bottom line they all come to in the end is that a computer screen is not "writing". There is no ink, no hand forming the letters, and no intention of permanentness. Old CRT monitors refreshed 60 times a second, so nothing on the screen had any permanence at all. Modern LCDs still have nothing physical - to quote Rav Ovadiah it's just "a combination of various lights".
As a side note, I'd like to point out that it's possible that e-ink (used in amazon kindle) is of a different status because it does contain ink, and is permanent when drawn. I haven't seen a rabbi (or anyone for that matter) address the issue of e-ink yet, so my answer does not apply to that technology. I'm not saying it's asur. I'm not saying it's mutar. I'm saying I don't know.