Your app should use a config file. This config file is loaded at runtime and shouldn't affect compiling. The allows users to download a binary and still use their own api key.
As Kornel says, you can include an example config file with a fake API Key, in your source control.
Another option, you could talk to the people running the webservices and ask for one of two things.
A temporary key, that only works for limited functionality. That would let users see basic functionality of your app, but some people would never update the key and just use the basic stuff.
Talk to the webservices to see if you they will give you a special API Key for your application. The open source version would require users to enter their own. But your binary could use a standard one.
The thought of using a config for api key's, isn't new or unheard of. Bit.ly services do it. And all the open source applications I see that provide use with Bit.ly ask for your username and api key before you can use it.
This is no different?
