Are there any examples of nuclear spin isomers having consequences for chemical reactivity?
Yes!

Papers in the field of preparation or selection of nuclear spin isomers have introductions that justify the work by potential applications and some of them include chemical reactions, either under that name or under the more descriptive terminology of “reactive collisions”. While no specific cases are cited as examples, the potential exist but only in limited situations such as bimolecular reaction in molecular beams.

From what I have seen, the canonical reference is “Detailed symmetry selection rules for reactive collisions”, Mol. Phys. 1977, 34, 477–504. This is the theoretical paper laying out the foundations of the effect of nuclear spin isomerism on reactions. I don't have access to the full text, but here is the abstract:

                   

It confirms what I suggested in my earlier comment: the different nuclear spin isomers have different rovibrational states, and accounting for effect changes the selection rules indicating which collisions are reactive or not.

One specific example which I found is “Observation of Ortho-Para H3+ Selection Rules in Plasma Chemistry”, Phys. Rev. Lett. 1997, 78, 3844–3847:


  Large dependences of the intensity ratios on the parent gases have been observed, indicating the spin memory of protons even after chemical reactions. The results clearly demonstrate the existence of selection rules on nuclear spin modifications in chemical reactions theoretically anticipated by M. Quack.


Their results are obtained on the formation of H3+ ions from a H2 plasma.