Does the mass of an object change as it moves away from the earth? The mass of a helium nucleus is less than the mass of two isolated protons and two isolated neutrons.  When the component hadrons are assembled, this mass is lost as energy ($E=mc^2$).  This makes it (at least sort of) seem like mass is a kind of potential energy.  Matter and antimatter can annihilate each other, releasing energy ($E=mc^2$).  This also makes it (at least sort of) seem like mass is a kind of potential energy.
If I have a mass at approximately sea-level on Earth, will flying it away from Earth increase its mass?  (If so, is this related to relativistic mass?)
EDIT:
The original question doesn't really ask what I mean to ask.  Suppose that, after the mass has been flown into space, it is stopped relative to Earth.  (I think that would require a geosynchronous orbit.)  Would its mass be different than it was before it left Earth?
 A: It depends on your reference frame. In the frame of the plate, its mass (strictly speaking, its mass-energy in its local freely falling frame, when you think about the problem in terms of general relativity) will be the same no matter what height it is at. However, for an observer located tens of thousands of kilometers above the Earth's surface, the mass-energy contribution of the plate to the total mass-energy of the Earth + plate system will be greater when the plate is elevated than when it is on the surface of the earth.
To lend credence to this answer, I direct you to Lubos Motl's response to a question of mine that is similar to this one: What is the mass of individual components in a gravitationally bound system?  Some of Lubos' most important explanations come in the comments to his answer. In particular: "the locally measured mass/energy in a gravitational field isn't the same thing as the contribution of this mass/energy to the total mass/energy as seen from infinity. Roughly speaking, the two quantities differ by the multiplicative constant $\sqrt{g_{00}}$, related to the gravitational potential..."
A: No. Mass is a kind of energy, but it is a different kind of energy than gravitational potential energy. As you move an object away from the earth, the gravitational potential energy changes, but the mass does not.
One caveat though: in the preceding paragraph, as in general, it is implicitly assumed that "mass" means "rest mass," namely the mass as measured at rest with respect to the object and very close to the object. If you make the same measurement when you are moving with respect to the object, or when you are reasonably far away, you may get a different result.
A: Yes. Suppose you have a plate at ground level. If you lift that plate up to a certain height, then the mass of the system Earth-plate is increased.
A: You can gain some insight for this problem by first considering the Einstein Equivalence principle. Imagine that the mass is at a height $r=R+h$ and move into a freely falling frame whose origin coincides with the mass. From your reference frame it simply looks like the mass is accelerating away from you at a rate of $g\,\text{m}/\text{s}^2$. Just before you hit the ground at $r=R$ and $t = \sqrt{\frac{2h}{g}}$ the mass is moving away from you with velocity $v= g\times\sqrt{\frac{2h}{g}}$ and thus according to special relativity the total mass-energy of the body is $$E = \frac{m_0c^2}{\sqrt{1 - \frac{2gh}{c^2}}}$$
Now let's expand this to first order in $\frac{gh}{c^2}$ to get $$E \approx m_0c^2\left(1 - \frac{gh}{c^2}\right) = m_0c^2 + m_0 g h$$
or that $$m_\text{rel} = m_0 + \frac{m_0 gh}{c^2}$$
We see that the mass-energy of the body is directly proportional to its height measured from the Earth. It looks like the Newtonian potential energy is contributing to the relativistic mass of the body, but this statement is incompatible with general relativity as the concept of gravitation potentials are non-existent. Rather the true reason for this difference is gravitational redshift. The intimate connection between energy and time evident in the definition of the four-momentum result in gravitational time dilation being responsible for the measured difference in energy. 
A: No.The mass of an object does never change no matter where it is in the universe.
A: Yes. If you take a mass and accelerate it it's mass will be its relativistic mass. But remember with relativity everything is "relative". If you say give a velocity (relative to earth) to a rocket, but you are on the rocket measuring its weight you'll measure the rest mass since the rocket is at rest relative to you. Relative to an observer on the ground thought he mass of the rocket will have changed.
A: People are mistaking mass for force. An object moving does not have more mass, it has inertia, and moving it away from the earth weakens the gravitational attraction between the earth and the object, causing a weight change. However, weight is a measurement of gravitational force on between two bodies containing mass. Mass, however, cannot and does not change. It can be added to or taken from by adding or removing matter, but even then, it is only located elsewhere. 
