What would be the maximum theoretically possible angular resolution? I have studied some basic astronomy, but I have difficulty conceptualizing the physics of luminosity and optics.  We use electromagnetic spectra to detect existence and properties of distant planets, but I am curious about the extent of what could theoretically be optically captured.  

(I am unsure if optical is the correct term, but I'm referring to what could be directly captured vs. computed.)

Assuming the use of technology far beyond what we currently are capable of, what is the maximum possible optical angular resolution we could theoretically obtain of a distant sky object?  Or, for example, of the surface topography of an outer planet/moon/dwarf planet?  

What would be the limiting factor at the extreme end of possible angular resolution?
The answer that you are looking for rests on two main pillars. Are you limited in your ability to resolve objects because of the atmosphere like most ground-based telescopes, or because of the fundamentals of the wave nature of light (as per space telescopes)? I am assuming the latter in your question. 

One of the biggest rule-of-thumb equations when it comes to astronomy is the equation that you are looking for in your answer. Astronomers define angular resolution with regard to the Rayleigh criterion: two point sources are regarded as just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other (Wikipedia). See Airy Disk functions and the Wikipedia article on Angular Resolution for a better idea as to what this means. 

The accurate, but still approximate, equation for the resolution of an object $\theta$ (radians), for a given wavelength of light $\lambda$, over some telescope's diameter $D$, is given below so long as the wavelength of light and the diameter is measured as the same length unit (i.e meters, feet, angstroms).

$$ \theta = \frac{1.22 \lambda}{D} $$

Obviously, the atmosphere is a pain, so, adaptive optics in most famous telescopes try and correct to this limit. To exceed this limit, major breakthroughs would need to happen in the wave nature of photons themselves. If your "technology far beyond" includes this, then, I don't think your question can be answered as you would hope aside from the standard "who knows". Currently, only space-based telescopes operate at this limit. 

For the most part, this is really all you need. It does not matter what object you are observing, this still always applies. 

Edited: Included unit specifications.