Possible to refinish hardwood floor without messing with quarter-round baseboard?
The answer depends on what you are willing to accept for a finished result.  Removing the quarter-round allows the edger to reach underneath what is visible when the quarter-round is re-installed. Even the most fastidious edging is going to be visible to close inspection if the trim isn't removed.  The extent to which it is obvious depends largely on the skill of the person doing the work, but it is impossible to hide completely unless the trim covers it.



Depending on the condition of the floor when you start, the condition of the trim, skill of the work, this can look decent or pretty bad:



Keep in mind that this isn't the only cosmetic consideration.  The other would be what the condition of the trim is.  If you have several layers of paint, it might look worse if you remove the quarter-round.  For example, the trim below would be a major pain to re-finish and look anywhere near decent.



My personal inclination would be to remove the quarter-round myself, have a professional do the floor sanding, and then re-trim it and paint it myself. I have quite a bit of experience installing trim, so your mileage my vary.