This isn't a Finalbuilder problem, but it took me half a day to figure it out, so I leave this here to help those who come after me.
In a solution file, the configuration 'Any CPU' is spelled with a space. In a project file, it has no space ('AnyCPU'), but msbuild knows to translate from space to non-space when it's building a solution file. I had a solution I was building but I needed to exclude a project from it, so I altered the build solution task to only build the selected projects, and selected all but one.
When you do that, Finalbuilder invokes msbuild on each of the selected projects, passing in the configuration ('Release|Any CPU' in my case). Because it's a project file, msbuild does not remove the space and you get a lot of errors of the form 'The OutputPath property is not set for project ... '
As I say, this is not Finalbuilder's fault and you can get the desired result by altering the build configuration in the solution to only build the projects you want.
For posterity's sake...
Nick