Luminous armor isn't a light spell
First, "[s]pell-like abilities cannot be used to counterspell, nor can they be counterspelled" (PH 180). Assuming something enables the drow to counterspell with a spell-like ability anyway, a drow that readies an action to counterspell then successfully identifies the foe's spell as the luminous armor spell still cannot counterspell it with the drow's darkness spell-like ability because the 2nd-level sanctified spell luminous armor [abjur] (Book of Exalted Deeds 102) lacks the descriptor light.
Likewise, the drow's darkness spell-like ability can't dispel the spell luminous armor once the foe casts it. The 2nd-level spell darkness [evoc] (Player's Handbook 216) and similar spell-like abilities simply have no power over the luminous armor spell absent it having the light descriptor.
What may happen
The luminous armor spell itself does have some power over spells with the darkness descriptor: it "counters darkness spells of 2nd level or lower with which it comes into contact" (BE 102). However, what's meant by this is unclear. That description, I think, both erroneously uses counters when it means dispels and erroneously italicizes the word darkness (i.e. like a spell or magic item) instead of leaving it untouched (i.e. like a descriptor). To be clear, countering magic already in effect isn't a thing, but dispelling magic in effect is; and other spells like daylight (see below) affect spells with the darkness descriptor of certain levels not, for example, spells with the word darkness in their names.
With that in mind, and using the 3rd-level Sor/Wiz spell daylight [evoc] (PH 216–17) that's referenced by the luminous armor spell as a guide, it seems reasonable to this DM to read the luminous armor spell as acting like the daylight spell with regard to affecting areas of magical darkness except automatically dispelling them instead of temporarily negating them. In other words, in this DM's campaigns, a creature wearing luminous armor dispels any magical darkness effect of 2nd-level or lower upon the luminous armor's wearer entering that area of magical darkness. Other DMs may rule differently, though, so it's a good idea to ask ahead.