Recently I received a letter from my credit card company asking me to send them a proof of identity/address as they have frozen my account and doing security checks. I can't be bothered to send them the documents they are asking for and the letter I received stated that they will close my account if I don't provide the required documents. I have no problem with them closing the account and I'll pay the entire amount I owe them in one go if they demand that. However the question is: would it look bad on my credit report?
Be very careful here. You should definitely phone the card company and confirm that they are the ones who sent you this letter. This could be a scam.
Update: Per the comments, always use a phone number or address that you have gotten from a verified source (back of your credit card or credit card statement) or from a trusted third party (directory information, ssl secured website, etc.) Though some phishing schemes can use decent fakes for websites, so be cautious when using the internet to research such things.
Having an account closed out due to security reasons will show on your credit report. Loan officers will see this. It creates a question in their mind, and they will likely ask you questions about it if you apply for credit in the future.
My advice is to contact your card company, confirm they sent the letter, and discuss what you can send them short of going through the process of renewing your passport. Some other combination of documentation. As long as you stay non-confrontational, they should work with you. Then, once you clear up the security question, you should evaluate whether you want to continue being their customer. On one hand, they're looking out for your interest, and you'll likely learn how to prevent this situation in the future. On the other, you are having to go through a bit of trouble. Balance that out and cleanly close the account if you feel you need to do so.
Your post didn't state what country you're in, so the answer to this will depend, but I'll offer some general advice anyway, since I think it is appropriate to the circumstance.
The statement "I can't be bothered" raises questions for me. After all, if you're worried about how they'll report it, why not just send them the documents they ask for to satisfy them and then just close the account at a later date like normal on your own?
If you don't want to run the risk of how they'll report this then MAKE the time to send them the documents they ask for? It's the easiest and safest course of course.