Does the property of being "virtual" for a particle depend on the observer? I've read at several places that a static magnetic (and electric for that matter) field can be thought of as made by virtual photons, at least that's what I understood.
Now, in Special Relativity we learn that an observer (in an inertial frame of reference) who sees only either an electric or a magnetic field will see both an $E$ and $B$ field if he starts moving. That is, he will see an electromagnetic field. And as far as I know an electromagnetic field is made of photons (not virtual photons).
Hence my question "is the property of being "virtual" for a particle (a photon in particular but any other particle is also welcome) dependent on the observer?"
 A: "Virtual" is not a property of a particle at all. And it is not true that pure electric and magnetic fields are "made out of virtual photons" or that a combined electromagnetic field is "made out of real photons".
A "virtual particle" is not a particle. It is an internal line in a Feynman diagram, which is in turn a graphic notation for a certain integral. No particle states are associated to such internal lines in the formalism. You may tell stories about them, but in the end, whatever they are, they are not particles in any sense.
Note that the accepted answer to the question you link says

From a field theory point of view, all static fields, whether electric, magnetic, the weak nuclear force or the strong nuclear force can be thought of as being mediated by virtual particles. [emphasis mine]

where now "mediated" does not mean that there are actual particles that "make up" the field, it means that the Feynman diagram between two charges in which a virtual photon is present gives rise to the electromagnetic force in the classical limit, see this question.
The relationship between the electromagnetic field (and with that also pure electric and magnetic fields), is complicated. The energy of the electromagnetic field corresponds to the number of photons in it, see this question, but this is a rather useless statement if the field state is not a number eigenstate. It is electromagnetic radiation, i.e. electromagnetic waves, which corresponds best to "being made out of photons", although also there the link is not clear cut, see this question.
A: The answer is actually simple and definitive. No. 
Indeed they have a mass (different from the respective "real" particle), but we know that $P^2=-m^2$, and since the square modulus of a 4-vector is observer independent, a virtual particle is observer independent.
(Curved space effect aside, like Unruh effect)
Still, by definition you cannot observe it, so...
