Not closing the output stream is designed for constructs like new TeeInputStream(..., System.out). I agree that making the close behaviour configurable is a good feature.
Instead of the auto-close feature, I'd rather make the option to close the associated output stream work in the close() method of the proxy stream. See the attached patch commons-io-TeeInputStream-close.patch for a proposed implementation.
One could use the AutoCloseInputStream decorator to get auto-close functionality:
new AutoCloseInputStream(new TeeInputStream(..., ..., true));