How did Joey & Chandler's Chick and Duck die?

They died, probably of old age, but it wasn't shown in the show. It has been revealed in the final episode, when Joey bought the Chick Jr. and Duck Jr. The group had lied to Joey that they had gone to a special farm where they were very happy but could not be visited.

From the script:

Joey: It's my house-warming present for Monica and Chandler.

Phoebe: It's a baby chick and duck!

Joey: Uh-huh. And I named them Chick Jr. and Duck Jr.

Phoebe: I did not see that coming.

Joey: Yeah, I figure they'll love it at the new house, you know? It has that big backyard. And then, when they get old, they can go to that special farm that Chandler took the other chick and duck to.

Phoebe: Yes.

Joey: Yeah. It's a shame people can't visit there.

Phoebe: That is the rule, though.

It seems likely that Chandler had them humanely destroyed. In the final episode there's a mention that they were taken to a special farm that can't be visited in order to live out the remainder of their lives.

Joey: Yeah, I figure they'll love it at the new house, you know? It has that big backyard. And then, when they get old, they can go to that special farm that Chandler took the other chick and duck to.

The average lifespan of a duck is 8-12 years and anything upwards of 20 years depending on the breed. The average lifespan of a well kept rooster can be anything between around 12-15 years so it seems highly unlikely either would be approaching old-age, even ignoring the possibility that they both coincidentally happened to die at the same time.

This whole gag is very strongly reminiscent of the 'Milner's Farm' joke from Season 1, episode 3 where Ross learns the fate of his beloved pet dog Chi-Chi.

The chick and the duck died of unknown reasons, likely age or eating something they shouldn't have. Joey was told they went to live on a farm because Joey is a giant man-child and can't be expected to handle the truth.

Rachel confirms they die in the last episode, without any stupid mention of being put down or euthanasia.

Joey: Sure, yeah. Will you just keep an eye on the chick [jr] and the duck [jr]?

Rachel: Chick and the duck? Didn't they die...

Phoebe: (interrupting) Dive. Yeah, they dove head-first into fun on the farm.

Chandler specifically kept the chick, and the duck, because he's against euthanasia. Instead of letting the chick be put down, he kept it, and ended up saving the duck, which was already mature/adult (not a duckling). He would never have them put down short of terminal illness or crippling age. He also wouldn't let Monica kill them, not that Monica would ever do that either. Neither of their characters are that heartless.

From Season 3 Episode 21, where the chick and the duck first appeared:

Chandler: If they can’t find a home for her, they kill her! And I’m not gonna let that happen to little Yasmine [the chick]!

The Duck is some age older than the Chick, as it was fully grown by the time Chandler saved it. This can explain how they could have died near the same time.

They were last seen in Season 6, and the last time they were mentioned but not seen on screen before the final was Season 7 Episode 2 "The One With Rachel’s Book":

Probably it's because they ate Rachel's face cream. It was in the episode The One with Rachel's Book. That was the last time they were mentioned and they were throwing up everywhere that is why Joey ended up sleeping in Rachel's bedroom. So I don't think Chandler killed them, it's just that maybe they were poisoned by the face cream they ate.

When they all went to England Phoebe did not feed Chick and Duck for four days probably that's the time they died. I didn't see them after that in any of the episodes.