Please test after each of the following steps that you haven't already tried. Stop when the problem is resolved. Back up all data before making any changes. Keep in mind that no one here represents Apple or can help with customer-service issues.
Step 1
A purchased app can only be updated by signing in to the App Store with the same Apple ID that was originally used to buy it. There's no way around that limitation, which also applies to free apps. If you can't sign in with the buyer's ID, delete the app and reinstall it. You'll have to pay for it again, if applicable.
Step 2
If you get the alert when trying to update a bundled iLife app, elect the Purchases page in the App Store and locate the app in your purchase history. If there's a button marked ACCEPT on the right, click it.
If you have a used Mac, the bundled apps were linked to the original owner's Apple ID and can't be transferred to you. Reportedly, customer service has issued redemption codes to some second owners who asked, but it's not guaranteed.
Step 3
If you're trying to update iLife or iWork apps that were installed from a purchased DVD, or if you have a refurbished Mac bought directly from Apple, contact App Store customer service for a redemption code. You may be asked for the part number of the DVD.
Step 4
From the App Store menu bar, select
Store ▹ View My Account
Enter your Apple ID password at the prompt. At the lower right corner of the window that opens, click the Reset button. Close the window.