Would an object accelerating because of gravity convert some of it's mass into energy? To change the phrasing, "Would an object lose mass when attracted by a gravitational field?" I'm talking about rest mass.
 A: Not really. A falling body converts gravitational potential energy to kinetic energy. That conversion does not affect the rest mass of the falling body, nor the rest mass of the body it's falling towards. It also doesn't affect the rest mass of the combined system of the two bodies.
However, this energy (whether in kinetic or potential form) does contribute to the rest mass of the combined system. Bear in mind that the rest mass of a system is the total energy of the system in its rest frame, and is not simply the sum of the rest masses of the system's components.
Consider an isolated system consisting of a pair of identical motionless spheres of radius $r$ and mass $m$ that are just touching, so the distance between their centres is $2r$. This system has gravitational potential energy
$$-\frac{Gm^2}{2r}$$
Now we use energy $E$ from some external energy source to separate the two spheres so that the distance between their centres is $d$, leaving the spheres at rest at the end of the process, so both spheres have zero kinetic energy in their rest frame.
Of course, the spheres will immediately start accelerating towards each other, converting potential to kinetic energy, but let's analyse the system's state while the KE is zero, to keep things simple.
The gravitational potential energy of the system has now increased to
$$-\frac{Gm^2}{d}$$
and by conservation of energy,
$$E=\frac{Gm^2}{2r}-\frac{Gm^2}{d}$$
This process does not change the rest mass of either sphere, but it does increase the total energy of the combined system by $E$, hence the rest mass of the system increases by $E/c^2$.

Note that this mass increase is very tiny! For example, if we calculate the mass equivalent of the potential energy of a 1 kg body raised 1 metre above the Earth's surface, which we can do using the Google calculator with
(1 kg)*(1 m)*(9.08665 m/s^2)/c^2
the result is $\approx 1.011 \times 10^{-16}$ kg.
