Cases where renormalisation isn't needed

Insertion of counterterms is needed, when you have certain non-vanishing diverging diagrams. For instance, the self-energy diagram in QED (Quantum Electrodynamics) for electrons leading to the mass renormalization, or polarization loop diagram for photon, which is logarithmically divergent, and renormalizes the charge.

One doesn't need counteterms definitely for the case of free theories, there is nothing to renormalize. It is a trivial example.

Less trivial case are the supersymmetric theories, where non-renormalization follows from the fact, that bosonic and fermionic contributions annihilate each other. One of the famous examples in non-renormalization of F-terms. Terms of form: $$ int d^4 theta W (Phi) $$ Where $W(Phi)$ is some superpotential of the superfield $Phi$ do no receive perturbative corrections. This can be proven via supergraphs, but the most beautiful argument due to N.Seiberg uses the power of holomorphy - https://arxiv.org/abs/hep-th/9408013v1.

There is a nice review by S.Weinberg - https://arxiv.org/abs/hep-th/9803099v1