I have a customer that is using us as a reseller they have the domain abc.com on our server and the name server ns1.abc.com and ns2.abc.com. They are going to be moving some of the sites including abc.com off our server onto a dedicated machine but want to keep there some clients on our server using ns1.abc.com and ns2.abc.com that they already use.

If the client uses ns1.123.com and ns2.123.com on there new server and updates abc.com to use these name servers will this interupt the service of the resellers customers on our server?

They can move the SITE abc.com without moving the DOMAIN abc.com. in that case just update "A" records of abc.com and www.abc.com maybe also mail.abc.com and MX entry, but don't change ns1.abc.com and ns2.abc.com.
The name servers can be anywhere...

If they want to transfer the DNS also, then they have to recreate all the zones (domains) at the new server, and also change the IP-s of the name servers at registar...