What kind of particles can be virtual? Only those in the table of the Standard Model? The cellebrated works of the teams of S. Savasta and R. J. Schoelkopf, showed that virtual particle can have effects which can be indirectly testable. See for instance the famous article
L. Garziano et al., "A Single Photon Can Simultaneously Excite Two or More Atoms", arXiv:1601.00886v2,
or,
R. Stassi et al., "Spontaneous Conversion from Virtual to Real Photons
in the Ultrastrong Coupling Regime", arXiv:1210.2367v2
(I recommend to read these articles very attentively for understanding how the virtual photons influenced the final result. In the 1st article, the presence of virtual photons in intermediate stages - see diagrams in fig. 4 - is testified by the frequency of the oscillations in figure 3. See the formulas (4), (5), (6). 
I recommend, stop reciting that virtual particles are only a tool in Feynman diagrams. Virtual particles can also appear in reality, in some experiments, but cannot be detected because of their too short life, and different problems as mass, and others. In the process described in the first reference, their presence violates energy the conservation - this is why one cannot detect these intermediate states, which are, of course, very short. Though, their presence can be deduced indirectly from calculi, as in the (excellent) work of Savasta's team - first reference.)
My problem is that in these works only virtual photons are generated. 
My question is which other types of virtual particles are known?
A colleague told me that only the particles in the table of the Standard Model can appear as virtual (of course, with abnormal mass or other abnormal features). Is that true? 
What about virtual protons, or virtual alpha-particles?
 A: Virtual particles can be any type of particle. A good example would be virtual pions. Nuclear forces can be modeled in a simple way through a one-pion exchange potential.
If virtual particles could only be elementary particles, that would be too good to be true. Then we would have a magic way of detecting structure at all scales, because composite particles would behave differently than elementary ones, at all energies.
A: Virtual particles are a mathematical construct in Feynman diagrams.
 They carry the quantum numbers of the named particle but are are off  mass shell, under an integration for  the process examined.
The virtual particle is a pole in the propagator under the integration from initial to final state.
I first learned field theory rules  from a nuclear physics model, with creation and annihilation  operators. If you can define a  consistent quantum field theory for any set of particles , those particles can also be virtual in the Feynman diagrams for calculating measurable quantities. One has to make sure though that the system is consistent for the mathematics to work.
