I run a Linux server with a number of domains. My open relay condition has actually been abused by spammers and I desire to close it, however I still need to send mail from my domain accounts. Will this cause any disturbances in my service?
It depends on what you understand by "closing open relay" or "open relay" in the first place.
Usually open relay is a condition where your SMTP server allows anyone with or without authentication, in or outside your local or trusted networks, with or without a valid email address to send email to ANY recipient, domain, address via your SMTP server.
In short words your server handles:
"from (ANYONE) to (ANYONE)"
Preventing this from happening is usually mandatory.
When you "close" the open relay you have to make sure that outside sources can still email to local recipients. Otherwise you wouldn't be able to accept incoming mail for your domain users.
Usually the preferred setup is to allow only authenticated users to send emails via your SMTP while accepting all emails for local domain/users.
from (AUTHENTICATED) to (ANYONE)
from (ANYONE) to (LOCAL ACCOUNTS/DOMAIN)
By closing an open relay, you are just ensuring that your mail server will only handle mail that is either originating from a user with an account on that server (even if it is hosting multiple domains) or is destined for a user on the system. All other mail delivery requests are ignored (the packets are dropped with optional logging).