The app works fine in the simulator, builds and runs.
When I attach my device (iPhone 3GS running iOS 4.3) and target the iPhone, the result is "Build failed" with "No issues".
Looking at the build results window, everything is green, has a checkmark, including the Build failed notice.
Expanding the transcripts show no errors.
I cannot exclude the possibility of a signing or provisioning error, but it gave me those errors, I ironed them out, and now it's not giving me any errors or issues, just "Build failed; No issues"
On one of the local Cocoa Heads mailing lists I was encouraged to attempt a small default sample project - something I should have done in the beginning to make sure my provisioning and signing worked. It built and ran on my device without issue, so all signs point toward a problem with my code or the project.
I'll be throwing things from the existing project into the new sample project, and if it fails in a similar way I should be able to narrow the problem down to a smaller region of interest.
I've added the main functionality of the app into a new project, and it's running fine. The only things I did differently this time was I did not set up a universal binary (iPhone only right now) and turned off all but portrait orientation. As far as I can tell everything else is the same, though if I decide to look into it further later I will diff the project files and see what else might be different.
So the solution for me was to start a new project a port all the functionality from the old project into the new project.
It could be that recreating the target, as Paul suggests in another answer, would fix it as well, but with such a simple project restarting from scratch was easy for me.