The answer depends on exactly what you're asking, but I think that you're conflating fields and states. In quantum theory (fields or otherwise), we have a Hilbert space $\mathcal{H}$ of states, and the quantum observables are self-adjoint linear operators on $\mathcal{H}$. You can think of $\mathcal{H}$ as being a space of wavefunctions. The superposition principle, as we usually know it, says that if $|\phi\rangle$ and $|\psi\rangle$ are states/wavefunctions, then so is $|\phi\rangle + |\psi\rangle$. This says that the space of states is a vector space, and is exactly why we take the space of states $\mathcal{H}$ to be a Hilbert space (as opposed to, say, some arbitrary set or topological space).
Now, if $\hat{f}$ and $\hat{g}$ are two observables, then of course $\hat{f} + \hat{g}$ is an observable. Observables obey this kind of "superposition principle" trivially, and this is not what we usually mean by the superposition principle.
To answer yes or no, I have to interpret your question. In quantum field theory, we still have a Hilbert space $\mathcal{H}$ (which is now a space of wave functionals), as well as quantum fields $\hat{\phi_i}(x)$, $i = 1, \ldots, N$, for each $x$ in our spacetime manifold. The quantum fields are observables, not states. 976
These are operators on $\mathcal{H}$, exactly as in quantum mechanics.
Interpretation 1. Do states in QFT obey the superposition principle? Answer: Yes, absolutely. The space of states is still a Hilbert space, and of course we are allowed to add two vectors in a Hilbert space to obtain another vector.
Interpretation 2. Does it make sense to add quantum fields? Answer: Yes, absolutely. The quantum field $\phi(x)$ at a point $x$ is just an operator on $\mathcal{H}$, and the sum of two linear operators is still a linear operator. For example, the operator $\phi_i(x) + \phi_j(y)$ applied to the vacuum would create a superposition $|i; x\rangle + |j;y \rangle$, where $|i;x\rangle$ is the state containing a single particle of species $i$ at the point $x$, and $|j;y\rangle$ is the states containing a single particle of species $j$ at the point $y$. (Technicality: these states won't be normalizable in general, but let's sweep that under the rug.)
Interpretation 3. If $\hat{\phi}_1(x)$ and $\hat{\phi}_2(x)$ are two quantum fields obeying the same set of field equations, then does $\hat{\phi}_1(x) + \hat{\phi}_2(x)$ obey the same set of equations? Answer: In general, of course not! But this is no different than in quantum mechanics. 669