Usually it just destroys the op-amp. The supporting circuitry should be fine, assuming it was designed properly to begin with. When a chip such as an op-amp burns up, the supply voltage (usually +/- 15V) may appear on pins that it otherwise would not have. This is where it can affect supporting circuitry. But in general circuitry is usually protected up to the supply rails through limiting impedance or, clamping techniques.