If the enzyme molecule is irreversibly inhibited, such as by covalent addition of the inhibitor to the active site, that enzyme molecule no longer can participate in the reaction with substrate. Thus, the effective concentration of enzyme in the solution is reduced. Since Vmax is directly proportional to the enzyme concentration, Vmax decreases. It actually decreases with time, since as more time goes by, more of the enzyme becomes inactivated by the inhibitor.

The common type of irreversible inhibitor that forms a covalent complex with the active site is competitive with the substrate. If you could measure the Km at some specific instant in the overall reaction in the presence of enzyme, substrate and inhibitor but before the covalent reaction occurs, the Km would be temporarily higher in the presence of the inhibitor than in its absence because the substrate would be in competition with the inhibitor prior to inactivation of the enzyme by the inhibitor.

On the other hand, if you separate in time the reaction of the enzyme with the inhibitor from the reaction of the enzyme with the substrate, such as by preincubating with the inhibitor then removing the excess, you would not see any effect of the inhibitor on the Km. That is because the portion of the enzyme that reacted with the inhibitor would have zero activity (lowering the Vmax, as explained above), and the portion of the enzyme that had not reacted with the inhibitor would have its normal kinetics.

It's possible to consider irreversible inhibition by an ideal noncompetitive inhibitor, reacting with the enzyme elsewhere than the active site. Since such noncompetitive inhibitors reduce Vmax without affecting Km, you would not see the temporary increase in Km described above for the irreversible competitive inhibitor when running the reaction with both inhibitor and substrate present.