Would removing heat from the compressor and adding heat to the turbine increase a gas turbine's efficiency?
Heat removal like that is sometimes done in gas turbine engines, not by cooling the compressor during operation, but by the use of an isobaric intercooler between the low and high pressure compressors.  These kinds of configurations are common in industrial gas turbines used for electric power generation, marine propulsion and other applications where extra weight is not as large of an issue as is with aviation gas turbines.

As far as adding this extracted heat energy back into the gas flow through the turbine section, this would not be possible as compressor outlet temps are in the neighborhood of 450°F where exhaust gases exit at around 1000°F, preventing a return of the compressor heat to the turbine section for extra power.  Typically intercooler exhaust is simply vented into the atmosphere.

More commonly, gas turbines are used in combined cycle powerplants which make use of the gas turbine exhaust heat to power a water boiler driving a steam turbine and secondary generator.  Some of the thermodynamic cycles used in these systems can be quite elaborate and can achieve an exergetic efficiency of over 60%, making them the most efficient heat engines on the planet.