When catching an arrow tipped with poison (like the ones used by yuan-ti) with an ability like the one the monk has or the one given by arrow catching gloves: does the ability reduce the poison damage? If so, is the total damage or the piercing one caused by the arrow only the damage that must be downed to 0 to catch the missile?
So, I'm not going to pretend it makes sense, but as far as RAW is concerned, you reduce the poison damage too, and you have to reduce the total damage to 0 in order to catch the arrow.
Deflect Missiles looks like this:
Deflect Missiles: Starting at 3rd level, you can use your reaction to deflect or catch the missile when you are hit by a ranged weapon attack. When you do so, the damage you take from the attack is reduced by 1d10 + your Dexterity modifier + your monk level. If you reduce the damage to 0, you can catch the missile if it is small enough for you to hold in that hand.
Deflect Missiles says that the damage you take from "the attack" is reduced. And the Yuan-ti's attack looks like this:
Shortbow. Ranged Weapon Attack: +3 to hit, range 80/320 ft., one target. Hit: 4 (1d6 + 1) piercing damage plus 7 (2d6) poison damage.
"The attack" means the whole attack, not some subset of the attack or just the weapon damage section of the attack. So Deflect Missiles will reduce the total damage you take from the attack, and you'll have to reduce the total damage of the attack to 0 in order to catch the arrow.
Like everything in D&D 5e, this is open to DM rulings in the name of realism, fun, or sanity. Personally, I'd think long and hard before I changed the way this works, though; depending on how you rule this you might end up making Monks immune to Rogues.
I don't have the ability or item text handy, but I can't imagine anyone trying to catch an arrow by intercepting the point with their hand. With that in mind, an arrow that is tipped with poison could be caught fairly 'safely' (if you screw up, you're no worse off than if you hadn't tried, right?).
If, on the other hand, you're looking for particularly nasty tactics, try using contact poisons on the shaft of the arrow. Those will get the target and/or the person who pulls the arrow out of the target for healing. (Imagine your cleric suddenly keeling over dead in the middle of trying to heal your combat injuries!)