The increase in gravitational potential energy plugs into E=mc2, as per Einstein...so the object increases it's mass by m=E/c2. It's tiny but there. Warming an object calorimetrically does the same sort of thing. Any increase in energy is always accompanied by a commensurate increase in mass. It's because the conversion factor is so small that it is generally left out of most discussions, but for completeness it must always be there.
As for the warping of Minkowski space-time, due to the separation of the masses there is always a commensurate change in the ambient neutrino sea flux when that occurs....that is also inseparable, and usually not included because the conversion factor is so small. pete.