"Sign in to download from the App Store" with a different Apple ID
Shane Hsu's answer is correct, but there is also another way to do it if you don't want to delete the app: There is one file for each app that contains the "receipt" for that app's previous download from the Mac App Store. The file is named receipt. Here is how to find it and move it to a different location, so that the Mac App Store will allow you to enter a different Apple ID:

Find the app in your /Applications directory:



Right-click, and choose "Show Package Contents":



In the new window that has opened, double-click the Contents folder to go into that folder.

You will now probably see a folder named _MASReceipt. "MAS" stands for "Mac App Store." It should have a file inside of it named receipt:



Move the _MASReceipt folder to the trash:



Now, close the Mac App Store and then launch it again. Hopefully, at this point, when you try to update the app, its Apple ID field will be empty.

If this doesn't work, you will probably want to put the _MASReceipt folder back, by opening the Trash: