Would an aircraft that uses propellers for movement but with a turbo jet like structure where the ignition chamber only heats the air a bit, similar to a hair dryer instead of releasing fuel and combusting it have any impact on the speed of the craft?
As a reference, consider the Meredith effect, which applies (applied) in the construction of the P-51 Mustang radiator.
From the linked Wiki site:
The Meredith effect occurs when air flowing through a duct is heated by a heat-exchanger or radiator containing a hot working fluid such as ethylene glycol. Typically the fluid is a coolant carrying waste heat from an internal combustion engine.1
For the effect to occur, the duct must be travelling at a significant speed with respect to the air. Air flowing into the duct meets drag resistance from the radiator surface and is compressed due to the ram air effect. As the air flows through the radiator it is heated, raising its temperature slightly and further increasing its volume. The hot, pressurised air then exits through the exhaust duct which is shaped to be convergent, i.e. to narrow towards the rear. This accelerates the air backwards and the reaction of this acceleration against the installation provides a small forward thrust. The air expands and decreases temperature as it passes along the duct, before emerging to join the external air flow. Thus, the three processes of an open Brayton cycle are achieved: compression, heat addition at constant pressure and expansion. The thrust obtainable depends upon the pressure ratio between the inside and outside of the duct and the temperature of the coolant.1 The higher boiling point of ethylene glycol compared to water allows the air to attain a higher temperature increasing the specific thrust.
If the generated thrust is less than the aerodynamic drag of the ducting and radiator, then the arrangement serves to reduce the net aerodynamic drag of the radiator installation. If the generated thrust exceeds the aerodynamic drag of the installation, then the entire assemblage contributes a net forward thrust to the vehicle.
It's important to note that the primary purpose of this construction was to counteract the drag of the exposed radiator. It's certainly unlikely to generate sufficient thrust to power an aircraft.
the amount of energy per second you would have to dump into the air passing through the "heating chamber" in order to make the turboprop engine work would have to be at least as great as that produced by the usual combustion of jet fuel in the heating chamber. Your homework is to do a quick energy balance on the system (using the two heat sources- jet fuel and electrical heat) to see what you would be up against.