How would exhaust heat recovery into the combustion chamber affect the efficiency of a jet engine?

Extracting heat from the exhaust will cool and compress it, slowing it down and reducing thrust. Recirculating the heat back into or downstream of combustion will raise the temperature at that point and on downstream to the extraction point. The inevitable thermodynamic losses mean that the returned heat does not quite compensate for the heat drawn out and overall efficiency will fall a little.

What does improve efficiency is to draw through additional air mass in order to increase the exhaust mass flow, even at the expense of lower overall velocity and temperature. This is what the bypass turbofan does.

A more radical modification is to swap the combustion chamber for a heat exchanger and heat the gas via hot fluid from an external source; the hotter the fluid the better. I have seen this proposed for the hydrogen turbopump in some versions at least of the SABRE air-breathing rocket engine. It was also proposed for nuclear-powered bombers in the 1950s, though I cannot recall if those included turbojets as well as ramjets.

Theoretically, in an infinitely efficient system, there would be no net difference. The energy lost in the exhaust would be added back to the combustion chamber to be lost again in the exhaust. Maybe, you could use this heat exchange side engine to run another system without losing engine power. But, there is never something for nothing. Although I've read that the cowled radiator for the piston engine spitfire did produce a small amount of thrust as a biproduct.

This is all predicated by the fact that the heat energy is dumped back into the whole system in or after the combustion chamber. If it's before the the compressors, it would lower the density of the air coming in. Your thrust is based on the mass of the air being accelerated. If it dumped into the compressors, that energy may be lost to the peak amount that the compressors can increase density and pressure. It would be lost in the current way compressors shed excess heat. If the compressors could be made of material that did not need to be protected from excess heat, then there would be no need for the heat exchanger adding heat. It would already be there as a biproduct of compression. Therefore, you would not need to rob it from the exhaust.

A couple of remarks:

1. The reaction jet engine produces thrust in the exhaust nozzle. In a subsonic engine the exhaust gas first accelerates in the convergent section of the nozzle then it expands in the divergent section. Expanding, the gas looses not only kinetic energy but it cools down as well. Since the thrust is developed in the last section of the nozzle then having a heat exchanger further down would actually reduce the thrust by reducing the ability of the gas to expand.
2. There is a reactive jet engine that uses heat exchanger. Is called Sabre and the developer tried to solve the problem of suborbital flight using one engine only. At takeoff the engine functions as a reaction jet. Once the speed exceeds the range of usability of the axial compressor, then a heat exchanger placed in the air intake cools down the air to maintain the efficiency of the axial compressor even at supersonic speeds. The airflow through the engine remains subsonic. However, on exhaust the afterburner provides the extra thrust required to maintain speeds in the vicinity of Mach5 to 7. This engine burns hydrogen and uses liquid helium as coolant, or so the developer says. This is a bold undertaking from a private developer.
3. To answer your question: The Sabre is an example of using a heat exchanger in the air intake. The only cooling in the exhaust is meant to keep the hot side operational and not to extract energy from exhaust gas because that will affect the thrust.