How important is the dimensional accuracy of filament relative to the detail of a print? In this answer user Barafu says,


  Yet I manage to keep my tolerances +- 0.05 mm which is enough for everything but miniature printing.


I have asked for clarification on that answer regarding what is meant by "miniature printing" but in the meantime, I want to ask the general question.

What impact does dimensional accuracy of filament have on final print quality, and why?  Does it vary between different filament types?
Dimensional accuracy is not as important as dimensional uniformity. I can print with undersized (or oversized) filament, adjusting the flow appropriately, provided the filament has a consistent diameter. When creating filament in-house, without expensive equipment, it is difficult to maintain the same diameter throughout the entire extrusion. It is likely this extrusion diameter (when creating filament, rather than the output of the actual print head) to which Barafu is referring when he mentions his tolerances: +/- 0.05&nbsp;mm in diameter. Which is reasonable.

The "miniature printing" comment likely refers to printing miniature models for tabletop gaming.

If the source filament becomes wider than expected, the output will have overflow, or more material than desired will be deposited, and this will certainly affect the quality of the piece.