One would assume that if air, or more precisely, oxygen gets into the reaction, it will shift, leaving the CO2 unreacted and converting the H2S into H2O and SO2. Does the reaction need to be run in an oxygen-free environment?
It is preferred, but not necessary. Oxygen would simply waste H2S the reaction can use to convert more CO2. In theory, oxygen would react and oxidize the H2S to form H2O and SO2, but this does not happen, because the temperature of the SWAP column is too low to initiate the H2S oxidation reaction.