Hello everyone,
I solved this riddle yesterday for my mom. She tried to update a couple apps but it asked for an email address that she no longer uses in association with her iTunes. The reason that it asks for a different iTunes account, rather than the one that you're now using as your main account, is because that particular app(s) was purchased using an the email address it is requesting. Every purchase you make is associated with a specific email, no matter if you now use a new account. Thunk of it like a credit card purchase, that purchase will always be associated with that card, none other.
Two options here:
1. delete and re-download these apps using your preffered apple ID, which will eliminate the need of remembering multiple IDs (this is a good idea if the apps are free, otherwise you'd have to repurchase.
2. Punch in the old password to update. If the account is no longer active, you're going to have a hard time retrieving the password if you've forgotten it.
I hope this helps.
Michael