How to build .sqlproj projects on a build server?

Answer: Microsoft now has an official NuGet package (see blog post).

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Old answer, prior to August 2016; provided in case the NuGet package doesn't work for you:

1. Install dacframework.msi (x86|x64)
2. Install SQLDOM.MSI (x86|x64)
3. Install SQLLS.MSI (x86|x64)
4. Install SQLSysClrTypes.msi (x86|x64)
5. Install SSDTBuildUtilities.msi (from the "Administrator Install Point" as setup in step 3 here)

Done!

Source: Headless MSBuild Support for SSDT (*.sqlproj) Projects.

Microsoft SQL Server Data Tools: http://msdn.microsoft.com/en-us/data/hh297027

Install the tools on build machine to fix the problem.