The purpose that Montresor had in telling Fortunato about the air in the vaults was to give him a false sense of security as they went deeper. Montresor wanted Fortunato to believe that he had his "friend's" best interest at heart so that when he ensnared him in the trap and began to bury him alive he would be completely stunned by the act.
This also acts as a bit of foreshadowing toward the future event because we know from our narrator that revenge is being sought, there is no friendship attempting to be forged here.
Montresor warns Fortunato about the air in the vaults in an attempt to lure him into a false sense of security by feigning concern for his health. In addition, by showing this concern he also creates in Fortunato a feeling that he is trying to avoid showing him the precious cask of Amontillado he has promised to show him; in other words, Montresor comes across to Fortunato as trying to "get out" of showing the prized Amontillado to him. Being the competitive individual that he is, Fortunato is determined not to be sidetracked from his destination, just as Montresor had hoped.