No, it does not.
The Drow Superior Darkvision and Sunlight Sensitivity abilities don't say anything about each other, so replacing Superior Darkvision does nothing to Sunlight Sensitivity. Similarly, Eyes of the Dark and Devil's Sight don't mention it either, so they have no effect (although Devil's Sight is better than standard Darkvision since it allows better vision in darkness than Darkvision and allows the character to see through magical darkness). Eyes of the Dark will have no effect on a Drow, since they already have Superior Darkvision, which is identical in all ways except name.
It is worthwhile to note that, technically speaking (and according to RAW), Sunlight Sensitivity has nothing to do with sun being in the character's eyes, since it applies
when you, the target of your attack, or whatever you are trying to perceive is in direct sunlight.
This means that a Drow character would suffer the effects of Sunlight Sensitivity simply by standing in sunlight, even if their eyes and whatever they're looking at are both in shade.
On the other hand, the intensity of any given light doesn't factor in, given that they have no listed weaknesses to bright light, light-based effects, or anything else that isn't direct sunlight. Presumably this means that Drow in 5e are assumed (setting permitting) to have some sort of hereditary magical curse granting them penalties in sunlight, rather than their eyes simply not being accustomed to bright light, but I don't know of any official sources that comment on the matter one way or the other.
That said, many DMs houserule things (or have a homebrew setting with different circumstances for the Drow) so that Drow simply have eyes that are vulnerable to bright sunlight and, therefore, simply need to shade their eyes somehow (and avoid looking at or attacking targets in bright sunlight). Talk to your DM to see how they plan on handling Sunlight Sensitivity and maybe they'll have an idea that works for you.