Effective field theories are families of non-renormalizable theories whose coefficients depend on multiple parameters, usually including an energy scale $E$ (or equivalent mass scale $M=E/c^2$) that, by construction, approximate the true underlying theory (of which they are an effective theory) at energies close to $E$.
Sometimes, $E$ is a nonphysical parameter, typically an additional scale introduced by a particular renormalization scheme - a necessity in theories like QCD; then $E$ is called a renormalization scale. In this case, all physical results must be (at infinite order) independent of $E$. Therefore the effective theories at different $E$ may be mapped onto each other (up to higher order terms that are neglected) by a renormalization group that also transforms the other parameters and the fields.
The parameter $E$ must not be confused with the cutoff $\Lambda$ in traditional renormalization schemes. The cutoff is a second energy scale that is moved to infinity to produce the renormalized quantities. The renormalization group describes the effects after this limit has been taken. In causal perturbation theory, there is no cutoff, as all quantities appearing in the theory are renormalized from the start. However, there may still be a renormalization scale, in which case there is also a renormalization group.
QED has a natural intrinsic scale, which is the energy corresponding to the electron mass. However, this scale is fixed by the physical constants defining QED. The cutoff introduces a second scale, which in traditional renormalization schemes is moved to infinity, while in causal perturbation theory it is absent from the start. In Scharf's causal perturbation approach to QED, this leaves no scale to be varied, hence there is no renormalization scale and no renormalization group.
One could, however, change the renormalization condition used by Scharf such that a renormalization scale is introduced; then a renormalization group would appear (indeed, in the 1995 edition of Scharf's book Finite Quantum Electrodynamics, this is treated in Section 4.8), and Scharf's original results correspond to its infrared fixed point. The latter is the appropriate scale for electromagnetic effects, which are visible at very low (e.g., everyday) energies.
Similarly for other field theories. For example, in theories with a Higgs sector, causal perturbation theory always produces the broken symmetry phase. Nonabelian gauge theories such as QCD, however, have no natural scale and hence always need a renormalization scale, even in causal perturbation theory. (Though they ultimately get one through dimensional transmutation; then the natural scale is given by the mass gap.)
Which operators are relevant or irrelevant just depends on the renormalization group structure, which in turn depends on the precise renormalization conditions imposed. Hence it appears in the same way in standard renormalized perturbation theory and in causal perturbation theory.