General relativity and quantum fields evolution in curved space There are many cases when we have to discuss the problem of evolution of quantum fields on GR background (inflaton evolution during inflation, axion field evolution etc). But GR isn't quantized as well at this moment, so we work only with classical evolution of fields. What to do if we need to take into account quantum effects? I saw few lectures where quantum fluctuations of inflaton during inflation was discussed, but I don't understand why it is possible.
 A: General relativity and quantum fields evolution in curved space
There's a problem with this I'm afraid. Gravity is all to do with curved spacetime, but that isn't curved space and curved time, it's a curvature of "the metric", or more simply, a curvature in your plot of measurements. See Baez: "Note: not the curvature of space, but of spacetime. The distinction is crucial." Also see this article where Einstein described a gravitational field as space that was "neither homogeneous nor isotropic". Then see http://iopscience.iop.org/0256-307X/25/5/014 : "It is pointed out that a curved spacetime is equivalent to an inhomogeneous vacuum for light propagation." Gravity is all about inhomogeneous space. For curved space, you want Percy Hammond: "We conclude that the field describes the curvature that characterizes the electromagnetic interaction". Google on electromagnetic geometry.  
There are many cases when we have to discuss the problem of evolution of quantum fields on GR background (inflaton evolution during inflation, axion field evolution etc). But GR isn't quantized as well at this moment
You can perhaps understand that by remembering that virtual photons are field quanta, and thinking of the electron and the proton attracting one another. They "exchange field" such that the hydrogen atom doesn't have much of a field left. However the same is not true when two hydrogen atoms attract one another, instead the fields are additive. At the root of this is the simple fact that the photon has a non-zero active gravitational mass, so a virtual photon is in some respects a virtual graviton.   
so we work only with classical evolution of fields. What to do if we need to take into account quantum effects? I saw few lectures where quantum fluctuations of inflaton during inflation was discussed, but I don't understand why it is possible.
Inflation remains hypothetical. I think it's superfluous myself, because some aspects of the universe can be likened to a black hole, and IMHO Oppenheimer's original "frozen star" is the correct interpretation. Then the infinite time dilation means there are no fluctuations. See Paul Steinhardt Disowns Inflation, the Theory He Helped Create.  
