There is some logic, and some luck.
The cap has but two functions; a) to keep things out of the bottle, and b) to keep things in the bottle.
In the case of EVO, there might be a few critters that would like to invade the bottle, to get at the EVO. If that is the case in your area, keep it covered.
I have not had too much problem with EVO evaporating, so my reasons for the lid are primarily related to the 'a)' comment; and, since I don't have a problem with critters, it's sort of mundane.
I have a capped bottle of dish soap on my counter. The problem I have is with the detergent drying and clogging the top of the pour spout, especially if there is a lid. Something that I haven't noticed with the EVO bottle.
The added benefit to the spout is the constriction of the opening; assisting me in restraint while pouring the EVO.