If a star merged with another star, the planetary system's survival depends on several variables, which include the angle at which the second star is entering, the speed at which it is traveling, and its mass in relation to the first star. A more satisfying answer is that the planetary system will never be the same, no matter what the second star is like. Binary star systems can contain planets, but they have a difficult time keeping these planets from ripping apart and from orbiting into the stars.
The planetary system will always survive, in the sense that the material of which the planets are made will not disappear--even if the merger rips the planets apart, their material will still be in orbit around the two stars, and although it will be a different orbit, the dust and rocks and gas can coalesce into new, different planets, if the new binary star system is stable enough.
The orbits of the planets, if they are completely not broken apart, will be vastly different, as they will have to adjust to the gravitational pulls of both stars.
Depending on what kind of a binary system is formed, planets may not be able to survive at all, although their material would stay in the area and probably spiral into the stars. If the gravitational pulls of the two stars cause extreme tidal forces, the planets may be ripped apart and the system may never be stable enough to reform.
A planetary system would probably not survive the supernova explosion of its star. The amount of radiation and the speeds at which this radiation is released from the star would destroy the planets. There would likely be no life on these planets when the star exploded, however, because the star would be a red giant right before it became a supernova, and all planets in the "habitable zone" of the system would have been engulfed by the star or heated up from its expansion.