Can Hartree-Fock mix more orbitals than there are electrons?
First some general considerations

Using more orbitals than electrons is not just possible, but almost essential for Hartree-Fock. Although in practice one can do calculations in a so called minimal basis set, this approach is lacking a lot of accuracy. Usually one adds additional basis functions, which leads to basis sets classified as double zeta (triple, quadruple and so on are also possible). 

For open shell systems even a minimal basis set includes more atomic orbitals than electrons are available. For example if you want to calculate a single carbon atom, you would have one basis functions for each of the $1s$, $2s$ and all three $2p$ orbitals. This gives a total of 5 atomic spatial orbitals (=10 spin orbitals), but only 6 electrons. (This is not really a good example, as occupying just one of the three $2p$ orbitals breaks the spherical symmetry of the atom, but I am leaving the scope of your question ...)

Virtual orbitals

Since all molecular orbitals (MOs) are expressed in this basis of atomic orbitals (AOs), it means we will get as many MOs as we put AOs in. If ordered by their energy, we first get a set of occupied MOs until all electrons are distributed. After that comes a set of virtual (or unoccupied) MOs. Those are still soluations of the Roothaan-Hall equations, i.e. they are eigenfunctions of the Fock-Operator, but they do not contribute Hartree-Fock energy.

However, virtual orbitals are important in correlation methods going beyond Hartree-Fock, e.g. Coupled Cluster or Pertubation Theory. Here the virtual orbitals become partially occupied and thus influence the ground state energy.

Equations

The ground state energy in case of closed-shell HF reads

\begin{equation}
  E_0 = 2\sum_a h_{aa} + \sum_a\sum_b 2J_{ab} - K_{ab}
\end{equation}

where the sum only goes over the occupied spatial orbitals, which equals half the number of electrons.
And for the Fock operator we have
\begin{equation}
  f(1) = h(1) + \sum_a 2J_a(1)-K_a(1)
\end{equation}
with the same restrictions to the summation. Since the Fock operator is applied to all MOs, this means the effect (coulomb repulsion) of the occupied orbitals is considered in each virtual orbital, but the virtual orbitals do not affect the occupied ones.

(Equations are taken from the book by Szabo and Ostlund, p. 134)