We report the detection of 23 OH+ 1 → 0 absorption, emission, or P-Cygni-shaped lines and CO(J = 9→8) emission lines in 18 Herschel-selected z = 2-6 starburst galaxies with the Atacama Large Millimeter/submillimeter Array and the NOrthern Extended Millimeter Array, taken as part of the Gas And Dust Over cosmic Time Galaxy Survey. We find that the CO(J = 9→8) luminosity is higher than expected based on the far-infrared luminosity when compared to nearby star-forming galaxies. Together with the strength of the OH+ emission components, this may suggest that shock excitation of warm, dense molecular gas is more prevalent in distant massive dusty starbursts than in nearby star-forming galaxies on average, perhaps due to an impact of galactic winds on the gas. OH+ absorption is found to be ubiquitous in massive high-redshift starbursts, and is detected toward 89% of the sample. The majority of the sample shows evidence for outflows or inflows based on the velocity shifts of the OH+ absorption/emission, with a comparable occurrence rate of both at the resolution of our observations. A small subsample appears to show outflow velocities in excess of their escape velocities. Thus, starburst-driven feedback appears to be important in the evolution of massive galaxies in their most active phases. We find a correlation between the OH+ absorption optical depth and the dust temperature, which may suggest that warmer starbursts are more compact and have higher cosmic-ray energy densities, leading to more efficient OH+ ion production. This is in agreement with a picture in which these high-redshift galaxies are "scaled-up" versions of the most intense nearby starbursts.