The error message you're getting there means that we were unable to connect to the sync server; we sent a message, but never received a response to that message. This could be a problem in OmniFocus, or it could be a problem with one the EDGE network the traffic is crossing.
I'm not sure why it only happens when the screen is turned off; if we take a look at the console messages your phone has logged, we may be able to figure out what's going on here, but that information isn't going to be particularly forum-post-friendly. Please contact the
support ninjas so we can figure out what's going on here.
One thing I'm trying to figure out is how to communicate to folks in threads like this that sync involves wirelessly contacting a server that is likely to be hundreds of miles away, fetching information from that server, doing some non-trivial processing of that information on the phone, and then sending information back.
Obviously, we want to make sync as fast as possible, and I understand that folks don't want to wait for syncs to complete. Especially on an EDGE network (with upload speeds comparable to a 56k modem), 30 or 40 seconds doesn't seem to be that long to wait to do all that. But that's because I know how it all works.
The previous two paragraphs aren't really about the specific problem you're having, Nicholas - it's just something I've been noticing and chewing on for a while. I think folks will always want syncs to take less time; when/if we hit a point where the network is really the limiting factor, I need to figure out how to communicate that without it sounding like a cop-out.
In any case, I've rambled a bit here - contact the support ninjas and we can pull the console logs off your phone. That will give us info on the specific traffic between your phone and the sync server; we'll use that info to figure out why sync is failing for you when the screen is turned off.