IntelliJ Idea, run code regardless of errors in unrelated project files

1. Edit the configuration.

2. With latest versions of IntelliJ, we need to remove the build from 'make before launch' option by clicking '-'. Below is the screenshot.

Modify your configuration to use Make, no error check option instead of Make.

For more information see this answer

Note that this may not be enough in your use case. Last resort is to use the solution provided by Vics answer.

As this answer suggests (https://stackoverflow.com/a/14793361/986160) you can use Settings > Build,Execution,Deployment > Compilers > Excludes and add the parts of the projects that are unrelated to be excluded from compilation. Make sure that the part you are trying to run doesn't have imports from those excludes.

Well, you can do it. The important part, of course, that the class with the main function compiles.

There are two ways you can go:

1. Compile the class that you want to run only (CTRL Shift F9). Run the main function with a right click. It will fail, but IntelliJ will automagically create a temporary running configuration for this main function. Now edit this configuration and uncheck the option to "Make before launch". Then go back to the file and launch the main as before.
2. Go to the configurations screen, then Defaults and then Application pane. Uncheck the "Make before launch" option and then you're good to go, as in section one, for any main function you'll encounter/create in this project.

Edit

This answer is outdated, and @ipandzic's answer should be used