I don't think they mean the same thing at all.
Does it feel like you're dying?
Here, "it" refers to a sensation of some sort, whether by internal or external stimulus.
Do you feel like you're dying?
This can refer to a state of being, either physical or mental.
Say you're in the hospital. You've been given a sedative. It makes you feel poorly.
"Does it feel like you're dying?" can refer to the effect of the sedative -- the implication is that the sedative and its effect in the body is the thing that is providing the person with the feeling as though they're dying.
"Do you feel like you're dying?" on the other hand makes no such implication -- here, the question is asking about the current state of miasma the person is experiencing, not about the effect the specific thing is having on them.
That you can ask this question out of the blue to a person and have it imply that the person could feel like death or dying does not necessarily mean that there are not times where you may choose to say one versus the other. It depends a bit on what context this comes under.
EDIT: OP requested examples.