Using a ferrite core transformer as an inductor for a boost converter
Will it work?


It will work as an inductor. However the amount of energy it can store will be a tiny fraction of that a proper inductor of the same physical size can store, because the core is ungapped, built for high permeability rather than high energy storage. You'll need to measure its value to see if it's suitable for your design. Measure the value at a range of currents, you may be surprised at how low the saturation current is. At the saturation current and above, the inductance collapses to a very tiny fraction of its low current value, which could cause a damaging current to flow in your driver.


Will there be a high voltage at the primary that is unconnected?


Yes, the voltage will be proportional to the number of turns on the winding. Having an open circuit high turns winding on the same core will reduce the SRF dramatically from what a single winding would give you. This may or may not cause you trouble at 71 kHz. Having only half the winding volume carrying your inductor current means higher losses than for a proper inductor.


The boost converter has a feedback. Will this damage the transformer?


Without a schematic, nobody will be able to tell you. I'm not sure what 'a feedback' is, or how you have it connected.


Will my MOSFET and diode get damaged?


Maybe. The on time of the 71.4 kHz signal could take the inductor current up beyond its saturation level, and so to very high indeed, which could damage your MOSFET, if your power supply has the current capability. When working with an unknown inductor, you should take baby steps, so measure the current while starting with a current-limited supply and using short pulses. You really need an oscilloscope for this. Note that using short pulses and observing the current with an oscilloscope is the same as 'measuring the inductance at a variety of currents'.