Could turbine or compressor stages of a jet engine be switched off to improve fuel efficiency?
You cannot "switch off" a stage of a turbine engine.

As seen elsewhere jet engines are a single block mechanically locked together: all the stages (compressor AND turbine) connected to the same shaft. If the shaft turns, all connected stages turn with it.

You can have designs with multiple shafts, but this increases the complexity (design and maintenance) heavily.

Even if we assume for a moment that you could have 1 shaft per stage, stopping one of these would be generally a bad idea.
For acoustic reasons the number of blades on the stationary part of the stage and those on the rotary part must be prime to each other (example: 37 blades on the stationary side and 39 on the rotary one); this means that when the rotary part stops you cannot align all the blades with the stationary ones, creating drag and losses within the engine, thus reducing the efficiency.




  How would efficiency of a Jet Engine change, if we assume for a second that its a possibility, to stop a few stages(turn them into free instead of locked)?


You would still decrease the efficiency: compressor stages are the ones actually pulling the aircraft forward (you would then produce less thrust*) and the turbine stages are the ones extracting energy from the exahust.

*: you could even risk having the flame going upstream if you stop too many compressor stages and not having enough pressure at the combustion chamber inlet.