In the PHB, when you apply poison on an arrow or a sword, the poison stays effective for 1 minute.

When a Drow hits someone with their crossbow, according to the Monster Manual, that person needs to make a save because of the Drow Poison.

But if you follow that drow for more than 1 minute, will the Drow poison lose its efficacy?

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.

No, because the drow is counted as an NPC, and therefore, follows different rules than a player character. The drow poison is not specified to be something that the drow must apply to their weaponry; it is an inherent part of the effects of the weaponry used by the drow.

We're looking at two different kinds of poison.

The basic poison described in the PHB deals an extra 1d4 poison damage to an affected creature, and it dries out and becomes useless after a minute.

The drow poison used by the drow in the Monster Manual doesn't deal extra damage: It inflicts the poisoned condition and, sometimes, the unconscious condition.

We know that it's called "drow poison" because it's in the Dungeon Master's Guide (page 256)—where it's described as having the same effects listed in the drow stats. There are bunch of other poisons listed in the same section, including purple worm poison, truth serum, and "burnt othur fumes."

Many of the DMG poisons are quite nasty, and none of them include any caveats about losing their potency after a set period of time. Basic poison, the only poison readily available enough to show up in the Player's Handbook, looks very tame in comparison—but a dose of basic poison costs just 100 gp, and a dose of drow poison is 200 gp.

In your specific case of Drow poison:

Yes, if everyone in your world follows the same laws of nature.

No, if the NPCs use different rules and exceptions than the players.

Chemicals react differently when exposed to air. It's up to the DM to decide how fast your chemicals dissipate. The DM is in control. :)

It's true that NPCs do not have to follow the same rules as the players, but me personally, like the NPCs in my world to follow the same rules as the players. It's great for consistency.