Since apps on iOS can access the clipboard should I worry about it when copying passwords from the keychain?
I do that when AutoFill feature doesn't work so I would not have to manually retype complicated passwords.
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No, but depending on your threat model, it may not be a problem.
There are a lot of ways an attacker could read the current and past contents of the clipboard, but usually this requires some kind of malware on your machine. At that point, you have arguably bigger problems, though it does expose you to greater risk.
So in short, if these are regular everyday use passwords, its not something to worry about unless you have a reason to be particularly paranoid. I wouldn't be copy-pasting nuclear launch codes for example. See also Is a password in the clipboard vulnerable to attacks?
As a side note, many popular password managers are vulnerable to memory-read attacks anyway, so even if you were careful about avoiding copy-pasting passwords, a sophisticated enough piece of malware could still steal your secrets.