The infinities in QFT occur at infinitely large momenta (infinitely short distances, known as UV divergences), and infinitely small momenta (infinitely long distances, known as IR or soft divergences). When you put the dynamics of the QFT in a box of finite size, and discretize the lattice inside the box for computations, you automatically do away with both the UV and IR divergences because the scales where they happen simply are not included in your model.
A common numerical QFT simulations is used for the simulation of quantum chromodynamics (QCD) in the low-energy regime, because this is not reachable by perturbative computations due to the specific energy behavior of QCD. In the most naive approach, you can view a discretized QFT as a large number of coupled quantum oscillators, where each oscillator corresponds to the value of the field of the given point. However, gauge theories such as QCD also require a treatment of the gauge fields, which are usually placed on the links between such oscillators. Most computations I have seen actually numerically computed the path integrals of the QCD rather than a Schrödinger-equation type of simulation. For more info see the Lattice QCD wikipedia page and the references given in there.