Can there be interference between a proton and an electron? For example, we know that we can interfere two different electrons or two different protons by employing them in a double-slit experiment. 
Now suppose, we mix protons and electrons and shoot them simultaneously for a double -slit experiment. Will the protons and electrons interfere with each other?
 A: It is a common misunderstanding that the double slit experiment is because you shoot electrons at the same time and they interfere.
In the double slit experiment, in reality, you shoot one single electron at a time. That electron acts as a wave on the fly, and its partial waves pass both slits at the same time, and these partial waves interfere with each other. The interference between the partial waves can be constructive, that is where you see a dot on the screen.
The interference can be destructive, that is where you see the dark areas on the screen.
The wavefunction of the electrons is the probability distribution of finding the electron at a certain position in all of space.
Now when the constructive interference happens, we use the phrase, the wavefunction of the electron collapses. In reality, nothing collapses, but the one single electron we just shot will make a dot in the screen, its wavefunction will reduce to a certain eigenvalue, that is a position on the screen.
To see an interference pattern on the screen, you need to shoot a lot of electrons after each other.
You are asking if you shoot an electron, then a proton, then an electron, then a proton, then will you see a pattern on the screen?
If you set up the electron pump (that shoots the electrons) so that each electron will have the same characteristics, then you will see a pattern for the electrons.
If you set up the proton pump (that shoots the protons) so that each proton will have the same characteristics, then you will see a pattern for the protons.
What you will see, is two patterns over each other on the screen. One pattern will be for the electrons, and another for the protons. But this will not mean that there is any interference between the electrons or the protons.
You are asking whether it is possible to shoot the protons and the electrons so from the same source that both type of particles will add to the same interference pattern and you could just see one interference pattern. This is not possible, since the wavefunction of the electrons and protons would have to be the same, and their quantum characteristics would have to be the same (it is not, electrons and protons are different quantum systems, with different quantum characteristics).
A: The root of this confusion is the visualization that is very commonly given to explain wave particle duality. When we say that particles behave like waves, we do not mean that in the sense of a wave in a puddle. In quantum mechanics (and field theory), we say we have an electron wavefunction (in QFT, we say we have an electron wave). This wavefunction, when expressed in the spacetime basis ($x,y,z,t$), gives us the probability of finding an electron in a region in a time interval. Similarly, the proton wave has its own wavefunction. 
But these two wavefunctions cannot be summed up to give us the joint probability distribution of the electron-proton pair. The waves in the puddle are representations of classical forces acting on a shared water surface and thus can be added (as vectors). But the electron wave and proton wave do not share any such surface. They are abstract representations of probability functions of two different quantum systems with their own characteristics and thus cannot be added that way --hence no interference.
To properly model the interaction between protons and electrons (or indeed two electrons), one needs Quantum Electrodynamics.
A: The protons and electrons will attract one another by the standard Coulomb force, and this will modify the interference patterns of both types of particle. However a proton wave does not interfere with an electron wave in the sense of wave interference known as superposition. The mathematics of this involves the apparatus of quantum mechanics, which I guess you are not familiar with or you would not be asking the question. The basic idea is that each particle has its own associated wave and this wave can have different parts which interfere (i.e. add and subtract) with each other, but the wave of one particle does not add or subtract with the wave of another particle. Meanwhile all these waves are affected by the local electromagnetic environment which they influence through their electric charge.
