Does an object gain mass as it moves further away from the Earth I have been reading about the mass defect in atomic nuclei recently and am trying to understand what it is that causes the defect.
To my understanding it is the loss of energy of the nucleus that causes an observed decrease in mass in the nucleus (using $E=mc^2$).
Therefore, when an object moves further from the Earth, since it gains gravitational potential energy, will it's observed mass increase?
Many thanks.
 A: Forget about E=m*c^2 for the system you desrcribe, because it is not a constant , it depends on velocity, whereas the mass of a nucleus is an invariant to Lorentz transformations.

I have been reading about the mass defect in atomic nuclei recently and am trying to understand what it is that causes the defect.

If you take a nucleus with a number of protons, $n_p$ and a number of neutrons , $n_n$  and weigh it, the mass of the Nucleus, $M_N$ < $n_p*m_p$ + $n_n*m_n$ . This is the mass defect, and is reflected the binding energy seen in this curve.

Therefore, when an object moves further from the Earth, since it gains gravitational potential energy, will it's observed mass increase?

This is a misunderstanding. A nucleus in deep space or in the sea  will still have the same invariant mass. It is only under extreme conditions that the masses can be affected, as for example in this study with very strong magnetic fields. Gravitational fields are orders of magnitude  weaker in comparison.
A: 
Therefore, when an object moves further from the Earth, since it gains gravitational potential energy, will it's observed mass increase?

No, but it is possible to change your analogy in order to make it correct. The thing that's analogous to the nucleus is the system consisting of both the object and the earth, O+E.
If an external source of energy brings the object farther away from the earth, then the total energy of the O+E system is increased, and by $E=mc^2$ this is equivalent to an increase in the mass of the O+E system.
A real-world example of this process, although in reverse, is in the black hole mergers that we observe in gravitational wave events. The black holes transfer a bunch of energy into gravitational waves, which take energy away into the outside world. Typically the system loses about 5-10% of its mass through the merger.
Note that if the system does not exchange energy with the environment, then the system's total energy is conserved, and its mass stays the same.
