What is the difference in performance between variable pitch prop and two fixed propellers for take-off and cruise speeds? Push-pull configuration aircrafts usually feature identical engines and propellers working only in tandem for multiple reasons like higher speed, better climb rate, reliability, cost, ease of maintenance, etc.

It may sound unpractical for piston engines were saving fuel will not outweigh the downside of one engine being a dead weight but may be a good idea for lighter electric or a combo, where engines may also be optimized for different propellers.

Can it be possibly beneficial to have tractor prop for take-off:


blown lift
no FOD risk
less noise outside


and pusher specifically optimized for cruising:


larger diameter as no ground clearance issue
no parasite drag
more stability
less noise in cabin


Benefits to consider:


efficiency
higher achievable speed
higher altitude
still two motors for redundancy


Problems:

Stowing nose prop.
Foldable prop should be cheaper than VP.

Less take-off power.
Electric motor for tractor could drive 4-bladed prop at peak power.
This doesn't specifically answer the question in the title, but from the body of your question I would say it's possible, if you had an electric push/pull configuration with a motor in the nose and one in the tail, or at the back of a fuselage pod with booms like a Cessna Skymaster.  

The aft facing motor could use simple fixed pitch folding propeller that folds aft and trails in the slipstream when not being driven (popular with motor gliders) and it would be used only for take off and climb, with climb-optimized pitch.  In cruise that motor could be shut off and its batteries assigned to the front motor and its blades would fold into a near zero drag trail position.  The front motor could have a fixed pitch prop optimized for cruise.

The key issue would still be worst-case climb performance.  It would have to be able to make enough thrust on the single cruise pitched prop at the front to climb decently if the back motor failed.  Possibly if the front electric motor had some sort of emergency boost feature, where it's torque could be temporarily increased (enough to get its coarse pitched propeller up to red line) for a certain number of minutes without overheating, would be sufficient for single engine safety, with the rear propeller folding back on its own for you.

For single engine performance on the rear motor, you might get away with just a brake for the front propeller to stop it from windmilling if its motor failed.  Not as good as feathering, but far less drag than windmilling.