Why should or shouldn't you kill a knocked opponent?

There are few reasons and/or strategies either way.

You should concentrate on the alive and shooting enemie(s) instead of killing off the defenceless ones. The knocked out opponents will still be there after your next fight, but if you start finishing them immediately, their friend(s) will come and kill you and you've lost the game.

Eventually you will acquire a gaming sense where you'd know if you can afford to kill a downed enemy based on the location of their friend(s).

You can also add the factor of strategy into this. You can pretend to be finishing off a downed enemy and quickly turn towards the location of their friend(s). They would assume you wouldn't be expecting them, as they can see the red bar of the knocked teammate going down fast.

You can also use a knocked-down opponent as bait. Generally, teams would try to help each other, so if you start finishing off a downed enemy, their friends might try to save them and get out of hiding.

TLDR;

Finish off a knocked opponent if you're safe from counter attack, otherwise always shoot the living first so that you don't lose the game by getting killed yourself.

A few patches ago, Fortnite added "siphoning" to every game mode. This means that when you kill an opponent, you get 50 health (or shields, if you're at full health) back once the kill registers. In duos and squads, it is not enough to knock the opponent for the "siphoning" effect to trigger. You actually need to kill them for it to count.

Because of this, more often than not, if you have a second or two during the battle to spare, it makes sense to kill the downed person. If you are being rushed, however, and have a lot of health and shield, it makes more sense to fight the living opponents first.

Here are my strategies:

If you are being rushed by an entire squad, focus on the squad attacking you unless you have bad weapons, low materials, or some other disadvantage. In that case, thirst the downed player to get that boost of either more materials, better weapons, etc.

If you think that you have a chance against the people attacking you, box the downed person so you know where they are and damage them a little to get his teammates restless and reckless. In this way, the other team - if you die - still has a timer to get to their knocked teammate. If they aren't fast enough (they do have to go through your box you built), you get the kill.

Also, if you are a solo-squad and getting rushed by three people, I would finish the person to get the kill and make sure the other team can't get a distraction to revive their teammate and make it a 4 vs 1.

tl;dr:

Finish the person if you need something or you are in a bad situation. Shoot the living people if you think you can kill them.