The Drow would still deal poison damage after a minute
Creature statblocks only do what they say they do. In the statblock for a Drow, they are listed as having a crossbow attack that targets must save against or be poisoned. There's no mention of poison needing to be actively applied to maintain this effect, or of the Drow even having a supply of poisons on-hand.
DMs can adjudicate the application of a Drow's poison, as a houserule/rule of fun
Personally, as DM, if I knew a creature's poison attacks were probably caused by a manually applied poison, I'd probably narrate that the poison is applied on either the first round of combat, or in an ambush scenario (which is traditionally where Drow show up in combat) they'd have applied it right before combat began. Alternatively, if the PCs were to get the drop on the Drow, it might be that the Drow have to choose between fighting without poison, or wasting their first turn applying poison.
One thing to note, however, is that the one minute limit on the poison duration only applies to Player Characters; it is not required to apply to NPCs of any kind. And as DM, you're not required to apply any kind of adjudication on this topic at all: if a statblock says the creature deals poison damage, then they deal poison damage, with no requirement to take into consideration other factors like time or circumstances.