Need Advice, Cannot Logon to Admin Site – Header Warning

Site Last Upgraded: 03/14/12
Issue: Website down and Header Links Dead
Error Message: Headers already sent

” Error Message: Warning: Cannot modify header information – headers already sent by (output started at /home/content/74/7620974/html/wp-content/themes/XXXXX/functions.php:1) in /home/content/74/7620974/html/wp-includes/pluggable.php on line 866 “

Help?! I am not sure what to do. I just started here, apparently my boss’ website has not been updated in over a year and the website designer has gone into retirement. I just checked it today and received the error message above. I then tried to logon to wp-admin and received the same message and do not have access.

From what I have read it could be several different issues, and I should try:
1. Deleting white spaces and adding code
2. Upgrade(especially since it hasn’t been done in over a year)
3. Delete last Upgrade and Restore Backup

I feel like this could be pretty serious.

Can you tell why this happened since nothing had been touched in so long? Based on the error message above, can you see what the issue is and where I should start? We don’t have very much money, but I am wondering if we should contact a WordPress Consultant.

I tried looking at the Files in the File Manager provided by the Host but not sure which file to start with and when I tried to look, based on the message, couldn’t see any issues with the ones I checked.

Thank you for your help.