Doppler effects are calculated based on the velocity of the sound source. Suppose the side of a swerving car (which is moving) grinds a guardrail (which is "not" moving). Will the screech sound higher-pitched to pedestrians ahead of the car than to those behind it?
The answer is yes and no. Part of the grinding noise will probably be local to the guard rail - for example the frequencies generated directly from deformation of the guard rail - these will not be Doppler-shifted. Another part of the grinding noise will probably be local to the car - for example the frequencies generated directly from deformation of the car - these will be Doppler-shifted. There will likely be a third noise source resulting from mutual deformation of the car and guard rail - these may (or may not) be Doppler-shifted depending on the local motion of the parts in question.
So the resultant grinding noise will likely have some components frequency shifted, but will almost definitely not have all components frequency-shifted (and the frequency-shifted components of the sound spectrum may not all be shifted by the same degree). So unlike the classic ambulance-siren example, in which the sound of a departing ambulance sounds like a slowed down recording of an approaching ambulance, the departing grinding sound won't exactly like a slowed down recording of an approaching grinding sound.
For an analogy of the first and second cases, imagine the car is driving past an enormously long harp consisting entirely of B-sharp strings! If the driver puts out her finger and runs it along the harp strings, each string will produce a pure un-shifted B-sharp note for the stationary listener to hear. If she is transporting an opera singer in the back seat and he simultaneously sings a long B-sharp note, the stationary listener will hear a horribly discordant B-flat.
The answer is yes. As a simple picture, imagine that the car hits the barrier at two points, A and B. Now imagine that at each point, the collision creates a single outgoing wavefront. To a person standing in front of the car, the wavefronts will be closer together because the wavefront from point A will have had more time to travel than the wavefront from point B. To a person standing behind the car, the wavefronts will be further apart for the same reason. The actual sound that you hear is just the combination of many wavefronts emitted from all the locations where the car encountered the barrier. This is the same mechanism that causes the doppler effect for a normal moving sound source.
EDIT: As Penguino pointed out, any sound due to sustained vibrations of the barrier will not be doppler shifted. However, if the dominant mechanism of sound production is the slip-stick effect due to the friction between the car and barrier, then the main point of sound production will travel with the point of contact. This sound will therefore be doppler shifted.