Accelerating away from a mass mitigates gravity's pull on the accelerating object. Would the same be true for an object accelerating towards the center of the mass?
No. It's not exactly the same.
When you're accelerating away from the massive object, the gravitational pull decreases - yes. But, you'll be under the influence of the field at whatever distance other than $\infty$. But in case of center of mass (assuming there's no other source nearby), you'll be pulled equally in all the directions (floating) which make you feel that you aren't affected by gravity. This is a different case. Because, you can be stretched into pieces if you're inside a sufficiently massive object.
So, it's better to pronounce this center of mass comically as a position of gravity balancing equilibrium..!
I'm not sure what you mean by:
because accelerating away from a mass increases the force you feel i.e. it make gravity feel stronger not weaker. You can do the experiment by going into an elevator. As it accelerates upwards and away from the centre of mass of the Earth you briefly feel heavier. As it accelerates downwards and towards the centre of the Earth you feel lighter.
I wonder if you meant that the gravitational field gets weaker as you move away from a heavy object ...
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and this is certainly true, and that speed is called the escape velocity. However this isn't accelerating because you're presumably thinking of the object being accelerated up to some velocity then left to move freely. The interesting thing is that if you're moving freely i.e. if you're not burning your rockets and nothing else is pushing you, then you will feel no gravity at all. It doesn't matter whether you're headed outwards, so gravity is slowing you, or headed inwards so gravity is accelerating you. In both cases you will feel no gravity.
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