Why can't you download app updates, what happens ? If you are getting an error message what does it say ?
For removing the card, how/when are you trying to do it ? You won't be able to remove when trying to download something, you would only be able to replace it. To see if you can remove it : Change or remove your Apple ID payment information in the iTunes Store - Apple Support
If that is what you are doing and if you aren't getting the 'none' option : Why can’t I select None when I edit my Apple ID payment information? - Apple Support