Why must titrate immediately the mixture of acidified iodate solution with kI?

Iodate ions are is strongly oxidising. They behave differently in different conditions.

2IO_3 \ ^- \ +\ 12H^+\ + \ 10e^- \rightleftharpoons\ I_2\ +\ 6H_2O

IO_3 \ ^- \ +\ 3H_2O\ +\ 6e^- \rightleftharpoons\ I^-\ +\ 6OH^-

If you don't make the titration fast, your Iodate will be reduced to iodine by water and with further be reduced by water to form iodide ions.