I skimmed that study, but it looks like all the samples they tested came from the same guy, and they didn't actually test putting people in MRIs.
Another group attempted to reproduce the results of Lee, et al by performing a similar in vitro experiment, but failed to detect any DNA damage.[1]
Nevertheless, in an article commentary published by the European Society of Cardiology [2], DNA damage was detected in an in vivo study of Cardiac MR (CMR) imaging, and a precautionary approach to MR imaging is recommended. They admit, though, that the study could not differentiate between the effects of the contrast agent used in CMR, and the effects MR imaging alone.