Similar issue, a little more detail and a different workaround in case it helps anyone.
Using the following:
iTunes 11.1.5
iPhone 5s with 7.1.1, iPhone 5 also with 7.1.1, and iPod nano running version 1.2
Not using iTunes Match on the Mac or devices.
Some songs that I purchased from iTunes quite a few years ago won't play completely through on any of these devices. On the iPhones the song keeps "playing" and the time moves advances normally but the sound completely goes away. On the iPod the song does the same thing for a few seconds but then skips ahead to the next one.
This was never a problem and all songs synced just fine on their own or as part of an album/playlist/etc. before iOS 7 so it's clearly an Apple iOS issue.
Tried syncing each device and unchecking music to remove everything. Synced again by checking music to add it back, no change.
Tried syncing again and checked 'manually manage' on the summary tab, opened iTunes, right clicked a couple of the problematic songs, ran a sync on one device, and the song worked normally.
Ran a sync on the other devices the standard way just by selecting them on the music tab (i.e. not doing a manually managed sync) and the songs that I added manually to the original device seem to be working across everything just fine.
The problem is that there is no way to tell which songs are problematic - nothing is grayed out, nothing has unique settings if you look at the info for each song, no visible difference at all.
So the only thing that seems to work is manually identifying problematic songs when you happen to come across them, then do a manual sync for just that song with a device, and it will subequently work.
I've seen a lot of other similar problems reported on forums but they all seem to involve using iTunes Match so this is a different deal. I'm HOPING that Apple support will eventually address the iTunes Match problem and that this will somehow also be fixed with it but I'm not holding my breath as they don't seem to be addressing that yet either, at least not in a timely manner.
In the meantime, at least I seem to be able to fix things one-by-one but that's going to get really old really fast especially not being able to tell which ones aren't going to work when I want to listen to them...