The name of DMRG is a bit misleading: the modern view is that it does not really have much to do with the idea of real-space renormalization (although that might be a motivation for Steve White to develop it, following the success of Wilson's numerical RG). DMRG is essentially a variational method based on matrix product state(MPS) representation of the ground state of 1D Hamiltonians, and any MPS with a finite bond dimension represents a state with finite correlation length. But CFT is gapless and has long-range correlations. In terms of entanglement, CFT has diverging entanglement entropy which MPS can not represent unless you take the bond dimension to infinity. Of course, in practice one can numerically approximate CFT ground state using MPS, and it works extremely well if the central charge is not too large. So practically DMRG is still the method of choice to study CFT in 1D systems in numerical simulations. However, conceptually these two are not naturally related. You may want to look at another technique, so-called MERA(multi-scale entanglement renormalization ansatz), which was designed to handle ground states like CFT which contain a lot of entanglement (more than area law). The connection to CFT there is quite explicit.