We present the first application of a new foreground removal pipeline to the current leading H I intensity mapping data set, obtained by the Green Bank Telescope (GBT). We study the 15- and 1-h-field data of the GBT observations previously presented in Mausui et al. and Switzer et al., covering about 41 deg2 at 0.6 < z < 1.0, for which cross-correlations may be measured with the galaxy distribution of the WiggleZ Dark Energy Survey. In the presented pipeline, we subtract the Galactic foreground continuum and the point-source contamination using an independent component analysis technique (FASTICA), and develop a Fourier-based optimal estimator to compute the temperature power spectrum of the intensity maps and cross-correlation with the galaxy survey data. We show that FASTICA is a reliable tool to subtract diffuse and point-source emission through the non-Gaussian nature of their probability distributions. The temperature power spectra of the intensity maps are dominated by instrumental noise on small scales which FASTICA, as a conservative subtraction technique of non-Gaussian signals, cannot mitigate. However, we determine similar GBT-WiggleZ cross-correlation measurements to those obtained by the singular value decomposition (SVD) method, and confirm that foreground subtraction with FASTICA is robust against 21 cm signal loss, as seen by the converged amplitude of these cross-correlation measurements. We conclude that SVD and FASTICA are complementary methods to investigate the foregrounds and noise systematics present in intensity mapping data sets.