This ever is a tough question.
If the character dies or something else happens with the character (disappearing in terms of getting lost) is up to the owner of this character. (This only applies if it is an original character. Canon characters may be given to other players.)
A good tactic (but still need a good roleplayer with a hand for it) is to let the character just disappear in the story. This does just mean to pull back from any conversation/event the character currently is engaged in and then just do as if the character is either not there or keep actions in the "out". This means if something happens (some actions are done) with this character another character just is talking about this happenings.
For example.
Tina and Tira are sisters (both played from different players). The player of Tina now decides to not go on but they both fought over the love of one lion. The player of Tina decided to stop playing directly in a quarrel between those two about their loved lion.
Tira could now push out Tina from being important in the story with a short lead out like:
Tira looked at the face of her sister. Something did break inside her. This was not her sister. She was only thinking of herself. Tira turned around without looking back.
'I will not lose to you sister. You will see..'Tira ran towards the den they once had together. She took her sleeping pelt and left the den right after. But where to go? Well what about having Mike take responsibility for her braking up with her sister. This could also be the opportunity she waited for....
In this example the player of Tira also takes advantage of the situation. Due the player of Tina will discontinue there will be no fight over the lion again. (Maybe!!!)
Later the sister can be mentioned in discussions or just in the story withouth taking over this character:
Mike:
"I haven't seen your sister lately. What happened to her?"Tira:
"Well you see... she needed some time for herself. She said I cling to her too much!"Obviously this was a lie. Tira did meat Tina this morning at the waterhole and they did not even greet each other. It seemed as if Tina had not forgiven Tiras bolt actions with Mike. But Tira could care less. It was not her fault that Tina never had been honest with her feelings. That was at least what Tira ever thought about her sister....
With this outcome you avoid playing this character but if needed you can explain why this character does not show itself anymore.
Over the next pages the sister Tina will most likely completely vanish from the story or just live in memories like "Tira remembered the times when she was a cub. Sometimes she missed her sister and the talks they had about guys...".
If you need another one to quarrel over the male lion here you can introduce another lioness. This may be played by a new player but has it's own story. It should not be too similar to the gone character.
To cut it short you should respect the wish of the player to not have the character being played, killed or getting lost.
But the player who leaves has not the right to forbid you to continue the plot without the character and completely let it vanish into the realms of forgotten characters.
If you can find no way around this character or to let it drop out like discribed above you could just write a small summery about the change of the past plot, introduce a new character that is close to the dropped out character and continue normal.
Example of our scenario with Tina and Tira.
Tina will drop out but you do not want to solve it like above but the plot needs the fight of two sisters (maybe the plot is all about this fight).
Here how you could solve it:
Plot-History-Change:
The sisters Tira and Nira are fighting over a lion in the pride named Mike.... (and so on)
You introduce a new character named Nira. (She takes the place of Tina, Tina will completely be gone off the plot. Every written conversation of Tina from before will be handled as if it happened with Nira. Nira should get her own profile and characteristics that may differ from Tina as well.)
Important here is that you do not play Nira as a Tina with another name but just as a completely different character that just has the same background as Tina. Like this you will not take over Tina and therefore not touch any of the character of the dropped out player.
(This should only be done if the player leaves on his own will. If the player does so he should not have the right to forbid you to exchange the characters. At least not if he is not writing an fade out of his character on his/her own to get it out of the story.)
The systematic is the same as if you would write a plot (new plot) that is about Simba and Nala but the daughter is not Kiara but a lioness names Mila that has a different characteristic but experienced the same (plays after Lion King 2).
As long as you do not really copy the other character you may take inspirations from other stories. (Everyone does this with the backgroundstory of the characters.)
Depending on how the player leaves (on good terms or bad terms) you may ask him what he would like the best (way one or way two). Also if you are not the storyteller (plot owner, creator of the plot) you should ask the storyteller for agreement to your way of pushing the dropping out character out of the story.
I hope I was able to answer your question.
Purring regards
Lodrill
EDIT:
Small addition about the comments above.
I would never kill a character if the player does not want to.
The character belongs to it's player and not to the storyteller (plot owner).
The storyteller can decide to kick someone out and write a lead out of the character (maybe the character gets banished).
The storyteller also may forbid actions (like magic, or killing off an important NPC or so) and edit where players are trying to force the story into a direction the storyteller does not want to.
But the only one who has the right to decide if a character dies or get lost is the owner of this character. (Even if the character gets badly injured is up to it's owner.)
Everything about the plot (story) is in the responsibility of the storyteller.
Everthing about the character is in the responisibility of the owner of the character.
Example:
A lion is walkind along a cliff. The storyteller may tell him that the stones under his feets are going loose and the way under his paws is falling down the cliff. But if the character really falls down the cliff, if the character will save himself by ramming his claws into the rock, if the character will run back and barely escape or if he falls down is up to the owner of the character. If it is important for the story the storyteller may decide that there is no other way than falling down the cliff. But how bad the character is injured is up to the owner of the character.
This rule is set for open roleplays like here where you have a story but the storyteller is one of the actors and not a mastermind that plays all NPCs.
(Open RolePlay, Forum Roleplay, Story Roleplay, Free Roleplay).
For PnP (Pen and Paper) roleplaygames like AD&D other rules apply due the storyteller (master) there is god and has the right to do anything with the characters. In PnP RPGs the storyteller does nothing else than telling the story, playing the NPCs and trying to make it as hard as possible for the actors (the characters). Also a PnP RPG ever has set characteristics shown in numbers (like life, mana, damage and so on) and a set fighting system.
We have a free RP here so no fighting system or rounds in fights do apply but everything is just determined by the players of the characters and the storyteller is just one of the players (if not for major plot decisions) and tries to lead the plot smoth and with the actions and descriptions of his characters.
Free RP:
Lina went through the rows of lions. It was her first hunt. She was excited....
PnP RPGs with set rules:
You see a lot of lions standing in a line to your right and your left. They are all watching you. The weather is....
So the way the storyteller discribes the situation is different and so are his rights to decide what happens to other characters.