The fight that's already won
The most important thing to check before you break off is whether the fight is already won for your team. If you engage in a 5v5 teamfight, drop one foe quickly, and deplete the health of another enemy who is forced to retreat, there are only three enemies left fighting. If most of your team is still healthy and you're in a position to chase, you should probably do so if no one else on your team does. If the fight is decisively won for your side, it's good to add some respawn timers to their team and some gold to your team by chasing.
If you do decide to break off, ask yourself whether it is realistic to try to chase. Chasing some characters is impossible. If you have to chase Kassadin for more than three or four seconds, you became a liability to your team because he will just use Riftwalk to escape. Chasing other characters might just be dangerous. Chasing Morgana, Ahri, or Lux is probably asking for a skillshot to the face and a spell combo afterwards. Chasing Teemo or Shaco is asking to run into traps. If your opponent runs more quickly than you, it's a good idea to stick with the fight. If your opponent has some death combo that can easily hit a fleeing opponent, save yourself the humiliation of losing 60% health from a full Ahri combo and just win the main engagement.
If a fight is already won for the enemy but one or two opponents are badly wounded, it can be tempting to chase and get something out of a bad situation. Don't do this. Your team probably wants to pull out of the fight, and the last thing you need is to die while chasing a kill. If two foes get zoned out and the remainder of your team is still fighting, you might be able to come back and win if you stick with the fight. If you can't, your team needs to back off and survive at all costs. Respawn timers and gold are the main factors in a game-ending fight, and if you die, you're giving the enemy team both of those things. If you're in a losing situation, it's important to cut the bleeding off and squeeze out advantage elsewhere. Don't try to chase a kill in some frantic two-minute chase that ends in a turret dive. Even if you get the kill, you're coming out behind.
Knight threatens queen
Zoning a critical opponent out of a fight early can win games. If you are playing a tank or bruiser, it's your job to disrupt the more dangerous foes, and if you can force the enemy carry out of the fight, your team is losing a small portion of its combat readiness while the enemy team is losing a ton of it. If you can threaten the enemy queen (ranged carry) with your knight (bruiser/tank/support), she has to disengage while your team's queens and rooks win the fight.
Frequently, you don't even have to be able to kill your target. Many "queens" will be able to zone you long enough that they can win the fight, and if you're a tank, you may not have a lot of options for damage. This might not matter. If you're Malphite and you opened the fight with Unstoppable Force, your job from that point on is to shut down the enemy carry. Your disc is entirely built around disabling carries, so if the carry runs, your options are to use your disc on bruisers or chase the carry. If you can zone the carry out while your team's DPS is allowed to work, chasing is a pretty good idea.
Almost all of this section is focused on dealing with ranged carries and not mages or other forms of DPS. Why not zone out a mage? The biggest reason is that there are lots of ways to hate on attack speed, and many of them are prevalent on tanks. The other is that a mage has probably burned all of her cooldowns already and won't be a threat for another 10-15 seconds. Attacking a melee DPS is probably a bad idea unless it's Master Yi or some other squishy character. Ranged carries contribute the most at the end of a teamfight if they're alive because all the big cooldowns have been used and only damage matters. If you can keep the enemy Vayne from picking up a quadra kill by keeping her from helping her team, good on you.
I can still fight!
Some characters can chase and still contribute to the main teamfight. The best examples of this are characters with global ultimates such as Karthus or Soraka. Other characters, such as Lee Sin or LeBlanc, can quickly return to the site of a battle after a brief chase.
In most cases, global ultimates are best used before
a teamfight, not afterwards. Ezreal's Trueshot Barrage can be shot back into a teamfight after he finishes chasing and killing a fleeing enemy, but Ezreal is a ranged carry and shouldn't be leaving a fight in the first place. He should use Trueshot Barrage early in the fight to hit as many foes as possible, weakening them before the fight starts. The same is true of Ashe's Enchanted Crystal Arrow. She could use it to kill a fleeing enemy, but this is a waste. ECA should be used early on in the fight to supplement the tank's initiation or to initiate the fight herself. Caitlyn's Ace in the Hole is a fairly good example of a good long-ranged chasing ultimate, since she doesn't really want to use it early in a fight (as a tank will block it).
Other characters are basically ultimate bots whose job in a fight is to deploy their ultimate. We mentioned Malphite earlier; there are a few others such as Nunu whose main duty in teamfights is to hit R. Once Kennen has spent his ultimate, his main contribution to the fight is done. He can still fight, but he's done the lion's share of his fighting in the first 4-5 seconds and can break off to do other things. If Zyra deploys plants and her ultimate into a fight, she's done her part and chasing an enemy is fairly reasonable. Sona is practically worthless in teamfights after she uses Crescendo, so if she can zone out a much more valuable foe, it's a good deal.
Don't. Just don't.
The rest of this week's Guidebook has explained some times when it's OK to chase. Most of the time it is not. When in doubt, don't chase.
Just don't do it! If you're ever unsure about pursuing a fleeing enemy, don't. If you stay and help the fight, your allies will love you and will give you +1 teamwork honor even if chasing was the smart play. If you see one ally break off to chase, stay back and help the fight and possibly even use V-click to tell Olaf that chasing is a bad idea. Although there are reasons to chase, the rule of thumb is simple: Unless your team has already won or you are basically worthless in the current fight, don't do it.
Good luck and have fun!
We understand what it's like to climb the skill ladder in League of Legends. The Summoner's Guidebook teaches you the tools you need to get a competitive edge. Whether you're climbing the ranked ladder, playing Draft Dominion, or getting crushed by intermediate bots, every enemy has a weakness. And every Thursday, Patrick Mackey shows how you can improve improve on yours.