The human eye on earth can only see about 8-10 miles (assuming a perfectly "flat" landscape) before the ground fades into the horizon. If we're talking about another planetary body, the minimum would most likely be the minimum radius for an object to be considered a planet, as you said planetary body.

If you eliminate that condition, there is a minimum mass that a body must have before its own gravity forces it into spherical shape, however this depends on multiple other factors, most significantly the material that the body is composed of. A quick google search revealed this relevant question and answer.

Obviously the body would have to be spherical (or near spherical) for us to see any curvature so then I would assume the question would become: how sensitive or adept is the human eye (or brain) at noticing a change in curvature over a long distance?

If you're trying to judge how straight a 12 ft.piece of timber compared to another, it shouldn't be so hard, but a much longer distance (miles) would surely be almost imposible.

Edit: phone autocorrects and repetition