DNA origami nanostructures are regarded as powerful and versatile vehicles for targeted drug delivery. So far, DNA origami-based drug delivery strategies mostly use intercalation of the therapeutic molecules between the base pairs of the DNA origami's double helices for drug loading. The binding of nonintercalating drugs to DNA origami nanostructures, however, is less studied. Therefore, in this work, we investigate the interaction of the drug methylene blue (MB) with different DNA origami nanostructures under conditions that result in minor groove binding. We observe a noticeable effect of DNA origami superstructure on the binding affinity of MB. In particular, non-B topologies as for instance found in designs using the square lattice with 10.67 bp/turn may result in reduced binding affinity because groove binding efficiency depends on groove dimensions. Also, mechanically flexible DNA origami shapes that are prone to structural fluctuations may exhibit reduced groove binding, even though they are based on the honeycomb lattice with 10.5 bp/turn. This can be attributed to the induction of transient over- and underwound DNA topologies by thermal fluctuations. These issues should thus be considered when designing DNA origami nanostructures for drug delivery applications that employ groove-binding drugs.