Actually the GR calculation is quite simple, at least approximately.
Photons redshift as they climb out of a gravitational well because they
lose energy. This means the higher observer actually sees things running
slower lower down, which is interpreted as time running faster for the
higher observer.
A photon of energy E has an effective gravitational mass of E/c^2, and
gravitational potential energy over height h in constant gravity field g
is m*g*h. So, the photon climbing in the Earth's surface gravitational
field of g = 9.8 m/s^2 will lose energy equal to (E/c^2) * g * h, or
E * (g/c^2) * h.
The fractional time change will be equal to the fractional energy
loss, and that is just (g/c^2) * h. g/c^2 is about 10^-14 using units
of meters. A commercial jet flying at 10000 m will therefore have a
speedup due to GR of about 10^-10. This is an order of magnitude more
than the relativistic slowdown of even the mile per second plane, so
it will swamp that effect in normal situations. Remember to set your
watch when you land.
This is a rough calculation; I'm not sure that the photon mass equivalence
can be plugged in directly in this way, but I think it is right to an
order of magnitude.
Hal