The near-resonant dressing of cold strontium gases and Bose-Einstein condensates contained in an optical dipole trap (ODT) with the 5s30s3S1 Rydberg state is investigated as a function of the effective two-photon Rabi frequency, detuning, and dressing time. The measurements demonstrate that a rapid decrease in the ground-state atom population in the ODT occurs even for weak dressing and when well detuned from resonance. This decrease is attributed to Rydberg atom excitation, which can lead to direct escape from the trap and to population of very long-lived 5s5p3P0,2 metastable states. The effects of interactions between Rydberg atoms, including those populated by blackbody radiation, are analyzed. The work has important implications when considering the use of Rydberg dressing to control the interactions between dressed ground-state atoms.