Maybe, probably not. All of the problems with the shuttle were not due to it being a spaceplane per se, but due to it being a sideways stack rather than a vertical one. Dream Chaser is designed instead to be on top of a rocket, either an Atlas V or Falcon 9.
Challenger failed because the failed o-ring between the segments of an SRB caused a jet of flame that impinged on the external tank. Falcon 9 doesn't use any SRBs. Atlas V doesn't use multi-segment shuttle style SRBs, and may not use SRBs at all for manned launches. Either way, that particular failure mode would be the fault of the booster and not the vehicle. In addition, by being on the top of the stack, if there is any sort of catastrophic failure of the booster, the vehicle is equipped with a launch escape system that was impossible on the shuttle.
The Columbia accident, as well as countless near-misses that could have resulted in a Columbia style accident, was due to debris detaching from the external tank and striking the orbiter. If the vehicle is on top of the stack, nothing that breaks off of the rocket can physically come into contact with the vehicle.
Therefore Dream Chaser isn't vulnerable to either of the causes of loss of a shuttle orbiter, and being a spaceplane has nothing to do with it.