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We present the data analysis pipeline, commissioning observations and initial results from the GREENBURST fast radio burst (FRB) detection system on the Robert C. Byrd Green Bank Telescope (GBT) previously described by Surnis et al. which uses the 21~cm receiver observing commensally with other projects. The pipeline makes use of a state-of-the-art deep learning classifier to winnow down the very large number of false positive single-pulse candidates that mostly result from radio frequency interference. In our observations totalling 156.5 days so far, we have detected individual pulses from 20 known radio pulsars which provide an excellent verification of the system performance. We also demonstrate, through blind injection analyses, that our pipeline is complete down to a signal-to-noise threshold of 12. Depending on the observing mode, this translates to peak flux sensitivities in the range 0.14--0.89~Jy. Although no FRBs have been detected to date, we have used our results to update the analysis of Lawrence et al. to constrain the FRB all-sky rate to be $1140^{+200}_{-180}$ per day above a peak flux density of 1~Jy. We also constrain the source count index $α=0.83\pm0.06$ which indicates that the source count distribution is substantially flatter than expected from a Euclidean distribution of standard candles (where $α=1.5$). We discuss this result in the context of the FRB redshift and luminosity distributions. Finally, we make predictions for detection rates with GREENBURST, as well as other ongoing and planned FRB experiments.