Feynman has tried to describe gravitation in term of spin-2 quantum field theory. A quite detailed account is given of this attempt in his "Lectures on Gravitation". However, my grasp of QFT is not enough to see where the theory fails to be satisfactory. To this day, the interpretation of graviton as a spin-2 particle is still something very speculative, and I wanted to know why, at least from a theoretical point of view. Any reference to specific published papers is welcome.
The problem lies in the fact that we define QFT perturbatively, and in doing so we meet divergences (as in almost any Feynmann diagram with loops). So, we need to remove divergences, and this process is called renormalization.
Without getting too much in the technicalities (if you need any clarification, I'll be glad to edit the answer), in order to renormalize the theory one must modify the action of the theory. This happens by redefining the coupling constants or adding other interaction terms between fields in the action, with new coupling constants to be found by experiment. In principle you could need to add infinite new terms, and infinite coupling constants: such a theory requires infinite measurements before giving predictions, so it is non predictive. For some theories, at low energy many coupling terms give a neglectable effect, so the theory can be used as an effective theory, but at high energies it gives no predictions.
Not with all theories the process of renormalization goes smoothly. In the standard model the renormalization can be always carried out by redefining already existing coupling constants, and no additional constants are needed: we say that the standard model is renormalizable, and it makes sense as a perturbative QFT. Sure, maybe it is inappropriate to use in strong coupling situations (like low energy QCD) due to its perturbative nature, but at least the theory is well defined. Note that the action of the standard model only contains a spin 0 boson (the Higgs field), spin $1/2$ fermions (leptons and quarks), and gauge spin 1 bosons (the vector mediators of the forces). No spin 2 particle is present.
Inserting the graviton has terrible consequences. The graviton is a spin 2 particle, and the renormalization process does not work for those particles. If you define gravity perturbatively (by starting from the Einstein-Hilbert action and linearizing gravity), the renormalization process does not go smoothly and you find out that you need to add infinite terms in the Lagrangian. I don't know if there is an energy scale under which this theory has only a finite number of important interaction terms, but it surely cannot be used as fundamental theory (that must be valid and predictive at all energies). So we say that quantum gravity is not renormalizable.
Feynman's theory is expressed as a perturbation series in the strength of the gravitational coupling. It will be qualitatively wrong in situations where gravity is strong (e.g., very high energy density), such as in the very early universe.