We present an algorithm for computing the Green function of the weighted biharmonic operator Δ|P′|−2Δ on the unit disc (with Dirichlet boundary conditions) for rational functions P. As an application, we show that if P is a Blaschke product with two zeros α1, α2 the Green function is positive if and only if |(α1−α2)/(1−{\bar α}1α2)|≤{2 \over 7}{\sqrt 10}, and also obtain an explicit formula for the Green function of the operator Δ|G|−2Δ, where G is the canonical zero-divisor of a finite zero set on the Bergman space.