Character A gives a target the dazed condition until the end of character A's next turn, but character A dies before their next turn.
What happens to the dazed condition? When does it last until now?
Your question is based on a misunderstanding.
Character A hasn't ceased to exist, they've simply gained the condition "dead". You can tell because the wording is the same as when a condition is applied, "character C is unconscious, character D is stunned, character E is dead".
Admittedly the "dead" condition isn't actually explained anywhere, and to my understanding this has never been errated, however I don't believe it says anywhere their initiative changes, or is somehow "removed" from the order, I'm not sure where you get that idea. If it does say that, you probably should have put it in your question.
In general, a character who is dead is also unconscious, which prevents actions, so people often skip them in the turn order, but this is not a rule. If they are dead and somehow still conscious, presumably they still get actions, but this doesn't change when conditions go away.
So Character A's turn still happens, and at the end of it the condition goes away, unless the power or the power's keywords specifically state it goes away earlier, such as on unconsciousness or death.
Generally, conditions and effects stay, however, specific conditions will go away.
Dazed has no such exception and stays until it has run its course. If the attacker is dead and the condition is End of Next Turn, then it stays until the attacker's turn in the initiative order comes up.
Duration: The effect that imposes a condition on a creature specifies how long the condition lasts. For instance, a cleric might use an attack power that makes its target blinded (save ends). However, some conditions have built-in rules for when they end. For instance, the prone condition ends on a character who stands up, and the dying condition ends on a character who regains hit points.
Things that go away when you're dead:
Things that work differently when the caster is dead:
Things that work normally when the caster is dead:
Conjurations and Zones explicitly say that they disappear when their caster dies:
Conjuration... Death Ends: A conjuration ends immediately if its creator dies.
Marked... A mark ends immediately when its creator dies or falls unconscious.
Grab...if you are affected by a condition that prevents you from taking opportunity actions (such as dazed, stunned, surprised, or unconscious), you immediately let go of a grabbed enemy.
Auras...Death or Unconsciousness Ends: A creature’s auras end immediately when it falls unconscious or dies.
Stance...A stance lasts until the character assumes another stance or until the character falls unconscious or dies
Zone... Death Ends: A zone ends immediately if its creator dies.
Anything that isn't a conjuration, a mark, a grab or an Aura does not end when the creature dies. However, specific conditions do require the user to be conscious to execute. For instance Dominated reads:
Dominated...The creature can’t take actions voluntarily. Instead, the dominator chooses a single action for the creature to take on the creature’s turn: a standard, a move, a minor, or a free action. The only powers and other game features that the dominator can make the creature use are ones that can be used at will, such as at-will powers. For example, anything that is limited to being used only once per encounter or once per day does not qualify.
Clearly the dominator must be conscious for this to have effect. However, the dominated would still be unable to take actions on their own.
However, these are the only (as far as I can find) conditions that end or change on unconsciousness or death. All other conditions remain until their durations end.
(All quotes are from the RC, I an add page numbers on request).
The rules on Zone powers explicitly state that the Zone ends when the caster dies. As other durations have no such explicit rule, I would recommend you leave the condition in place until it would have normally ended.
It should be obvious, but I'll mention it anyway: if the effect requires the caster use an action to sustain it, the effect is going to go away (when that character's turn would normally be, going by the above, unless it's a zone being sustained), as the caster can no longer take the required sustain action.
Short version: There is no actual hard answer on that question.
Okay lets say player didn't die instantly. He's just at 0 hp's and trying to stabilize via saving throws. In that case his powers still take place. Since rules says:
Dying: When and adventurer's hit points drop to 0 or fewer, he or she falls unconscious and is dying
He cant take new actions and with unconscious condition which means he's helpless(grants advantage), can't take actions, -5 to all defences etc. Since he can't take actions, he can't sustain some powers, but nothing says about powers being negated. So in that case all 'until the end of your next turn' and 'until the end of encounter' powers are still in place.
But if player was killed instantly
Death: When an adventurer takes damage that reduces his or her current hit points to his or her bloodied value expressed as a negative number, the adventurer dies.
Thats all that says in rules. The rest is house ruled most of the times.
In my group we're trying determine effect and think if it's logical to continue. Warlord's inspirations can't work without living warlord. But somekind of bleeding effect can continue without creature that caused it.
Although between wizard's community there is an idea that all powers that don't need to be sustaned(via minor action) are lasts as intended. Even if source dies. Magic. 4 ED is full of illogical things and rules that are officialy confurmed by WOtC. I will not be surprised if this is the same case.
That's all I can say about this situation.
I would use common sense and say in general it would last until the end of Player's A next turn would occur. The only time I would think it would end earlier is if Player A needed to maintain some type of concentration (e.g., spell or feat) to maintain the effect. If the dazed condition was due to injury or stun effect, it wouldn't be affected by Player A's condition.