A Dyson-renormalizable QFT is defined as one where you can renormalize the theory by renormalizing a finite number of parameters (in 1+3-dim spacetime usually the wave-function normalization, masses and coupling constants) loop-order by loop-order of perturbation theory. Usually you have an infinite number of divergent Feynman diagrams symbolizing proper vertex functions. E.g., in simple ##\phi^4## theory the superficial degree of divergence of a diagram is ##4-E## (where ##E## is the number of external (truncated) legs of any 1PI truncated diagram). Due to field-reflection symmetry the 1PI diagrams with an odd number of external legs vanish. Thus the only divergent diagrams are the self-energy diagrams ##E=2##, which are renormalized by wave-function and mass renormalization and the four-point proper vertex function, which is renormalized by the coupling-constant renormalization.
It has been shown by Bogoliubov, Parasiuk, Hepp, and Zimmermann that superficially renormalizable theories are indeed renormalizable. When it comes to theories with symmetries there's, however, a bit more to it. Usually symmetries restrict the freedom of terms in the Lagrangian. E.g., in QED if you want a superficially renormalizable theory you cannot write down a four-photon interaction contribution since there is no gauge-invariant term which has a coupling constant with non-negative energy dimension, i.e., there is no such term that is superficially renormalizable. On the other hand symmetries, particularly local gauge symmetries, lead to strong constraints on the proper vertex functions themselves known as Ward-Takahashi or Slavnov-Taylor identities, and these can help to make superficially divergent contributions in fact finite. In QED the four-photon vertex is superficially logarithmically divergent, but the WTI of this function in fact tells you that it delivers only a finite contribution.
Now another large class of important QFTs are the socalled effective QFTs. They are non-renormalizable in the sense that you need an infinite number of parameters to renormalize the theory. These theories are also based on symmetry like chiral symmetry as a guideline to build effective QFTs for hadrons, which are, as composite particles, only effectively described as elementary quantum fields for low collision energies, at scales where you don't probe their composite structure too much. Thus you have an expansion in powers of energy and momentum (usually relative to some scale, which in chiral perturbation theory is ##4\pi f_{\pi} \simeq 1 \; \mathrm{GeV}##). Then you start with a tree-level Lagrangian obeying the symmetries containing all terms not only the Dyson-renormalizable ones allowed by the symmetry. Then you start at a certain order in momentum. Usually the more loops are contained in a proper-vertex diagram the higher the momentum order of divergent pieces get, and you have to renormalize the contribution by renormalizing the corresponding coupling constants in the Lagrangian. One can show that this is possible, i.e., usually you are not forced to introduce symmetry-violating types of contribution to the diagram.
However, there are important exceptions, known as anomalies. An anomaly occurs when a classical field theory obeys some symmetry which, however, is necessarily destroyed when quantizing the theory. A famous example is the axial anomaly. When you consider massless QCD it obeys a symmetry where the quark fields are multiplied by a factor ##\exp(-\mathrm{i} \alpha_A \gamma^5)##. Now it turns out that you cannot renormalize the theory such that both the vector currrent (due to the usual symmetry by just mutliplying the quark fields with a usual phase factor) and the axial current stay conserved. The vector current, however, must be conserved. So you necessarily have to break the axial U(1) symmetry. An that's in fact a good thing, because it explains the decay of neutral pions to two photons.
