Assume my familiar is afflicted with poison (the poison status is just an example). On my action, before the familiar takes further poison damage, I dismiss it.
Does the status wear off while dismissed, or does it die in the pocket dimension?
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While the PHB does not specify this explicitly and thus the decision on this is entirely up to your DM, it is reasonable to assume that a dismissed familiar is not put into a state that freezes status effects on it. If your familiar was afflicted with a status effect when you dismissed it, I see no reason why that status effect wouldn't continue to impact it.
The text of the spell simply specifies that it
disappears into a pocket dimension where it awaits your summons.
Other spells that send things to other planes (like Banishment) don't put them into suspended animation, time flows normally while in a Demiplane, and so on.
We do have one example of a way to suspend ongoing Poison or Disease damage, and that's to Petrify a target.
[...] a poison or disease already in its system is suspended, not neutralized
We also have Sequester that puts a target into a state of suspended animation (though there is some debate as to whether or not that suspends the effects of ongoing damage). Find Familiar doesn't have any language similar to these two effects.
So, in short, while there are other things in the PHB that imply or outright state that the target of those effects would have ongoing damage suspended, there is no language in Find Familiar that implies this to be the case for your familiar. It's literally just hopping off into a little pocket world where it waits for you to summon it back.
Sorry, your Familiar probably died.
This is ultimately up to your DM, but it seems like your DM might very plausibly rule that your familiar keeps taking poison damage.
There's good news. The find familiar spell says:
When the familiar drops to 0 hit points, it disappears, leaving behind no physical form. It reappears after you cast this spell again.
A strict Rules As Written reading of this text suggests that it might be technically possible for your familiar to die, if it gets affected by Power Word Kill or some other effect that kills it without reducing its hit points to zero.
A less strict reading might suggest that your familiar is a magical effect, and casting this spell brings it back even if it was destroyed.
Regardless, there's no way that getting hit with poison damage can "kill" your familiar. You can cast the spell again and bring it right back.