We show that major existing Monte Carlo burnup codes are numerically unstable in cycle calculations of critical reactors; spatial oscillations of the neutron flux can be observed even when relatively small time steps are used. This is caused by using the explicit Euler or midpoint method that appear to be numerically unstable with the step sizes common in cycle calculations. More stable methods that are common in deterministic burnup calculations, like the modified Euler method, can easily be introduced into the Monte Carlo burnup codes.