To make sure I'm not comparing apples and pears, my (attempt to) answer the question will be broken into two parts: comparison of single-stranded nucleic acids and double stranded ones.
Single stranded DNA and RNA
Both DNA and RNA can form single-stranded complex tertiary structures in which the secondary structure elements are associated through van der Waals contacts and hydrogen bonds. The presence of a 2'-hydroxyl group makes ribose ring prefer different conformations than deoxyribose in DNA. Also, since 2′-OH moiety is both a hydrogen donor and acceptor, it provides RNA with greater flexibility to form 3D complex structures and stability to remain in one of these conformations. As Aleadam notices, this paper shows that tRNA and its DNA analog form similar tertiary structures though tDNA is not as stable as tRNA:
Therefore, we submit that the global conformation of nucleic acids is primarily dictated by the interaction of purine and pyrimidine bases with atoms and functional groups common to both RNA and DNA. In this view the 2-hydroxyl group, in tRNA at least, is an auxiliary structural feature whose role is limited to fostering local interactions, which increase the stability of a given conformation.
These authors also show that at least one loop in the tDNA analog is more susceptible to cleavage by a restriction endonuclease. In this region the tRNA has a water molecule hydrogen bonded to 2'hydroxyl group.
I was not able to find more of such interesting comparisons in the literature.
Double stranded DNA and RNA
Both DNA and RNA can form double-stranded structures. Again, sugar conformation determines the shape of the helix: for DNA helix it's usually B-form, whereas helical RNA forms A-geometry under nearly all conditions. In RNA helix we find the ribose predominantly in the C3’- endo conformation, as 2'-OH stericly disfavors the C2'-endo conformaion, necessary for B-form geometry.
dsRNA and ssDNA often provide a signal to the cell that something is wrong. dsRNA is of course seen in normal processes like RNA interference but it can also stop protein synthesis and signal viral infections (cf. double stranded RNA viruses). Similarly, ssDNA is much more prone to degradation than dsDNA, it often signals damage of DNA, or infections from single stranded DNA viruses and induces cell death. Therefore, due to their functions, under normal conditions DNA 3D structure is mostly a double-stranded helix, whereas RNA has a single stranded, "protein-like", complex 3D structure.