Import/Export TeamCity build configuration from one server to another

Unfortunately there is no such thing. TeamCity 8 made the situation a little bit better though by introducing a Build Id format (project name + build config name, can be overwritten) that makes it feasible to "hand copy" build configurations:

Basically under the hood all your TeamCity build configurations are really just XML files in the BuildServer\config\projects\ folder and sub folders. While I haven't tried this you should be able to just copy your project folder or build config XML to the appropriate destination on your new TeamCity instance if the ids don't collide. At the very least you can definitely overwrite existing projects with updates this way (something I have done in the past to dynamically change build configs "on the fly").

Of course if your build config depends on other builds / artifacts those ids have to match as well, so either you have to copy those as well or adjust the ids accordingly. Same goes for agent requirements.

Edit:

With TeamCity 9 out now there's a much better option to move projects between TeamCity servers built in:

Now TeamCity provides the ability to move projects among servers: you can transfer projects with all their data (settings, builds and changes history, etc.) and with your TeamCity user accounts from one server to another. All you need to do is create a usual backup file on the source TeamCity server containing the projects to be imported, put the backup file into the /import directory on the target server and follow the import steps on the Administration | Projects Import page.

For a full summary see what's new in TeamCity 9.

For TeamCity 9 and above:

1. Make sure both instances of TeamCity are running the same version.
2. Export data from TeamCity: Using the web UI on the source machine, go to Administration -> Backup and do a Basic backup. It will tell you the path to the backup file created.
3. Import data to TeamCity:
   1. On the target server, open the web UI and navigate to Administration -> Projects Import. This will tell you the path to the import directory.
   2. Copy the backup file to the import directory, refresh the web UI, and click 'Configure Import Scope'
   3. Select the projects and categories of data you want to import. Given that the question was just about build configurations, you would uncheck importing users and groups. Click 'Start import'.

TeamCity 9 has this ability builtin - https://confluence.jetbrains.com/display/TCD9/Projects+Import