Just like larrylampco above, I am assuming you have sites hosted on a dedicated server with company A and you would like to switch your sites to a server with company B. I am also assuming you are currently hosting your own DNS on your own domain on your dedicated server with Company A (such as ns1.yourdomain.com and ns2.yourdomain.com) and not using provided nameservers (such as ns1.webhostname.com and ns2.webhostname.com) which are mostly used when using shared or reseller hosting.
If both servers are using cPanel then you can create a DNS cluster setting the sync option to "Synchronize Changes". This should sync all host files from server A to server B. At this point server B would have all the information on it to start serving DNS queries (even though we haven't started sending any yet).
Next would be to edit the host file of the domain you use for nameservers to account for the change of nameserver IP. So log back into WHM on server B and go to "Home »
DNS Functions » Edit DNS Zone" and select the host file for the domain you use for nameservers. Then change the A records for your NS1 and NS2 and point them to server B.
At this point you will have to log into your domain registrar and register the nameservers with the same IPs as used in the paragraph above, this is to ensure proper GLUE. This is because your nameservers are on the same domain you will be using for content. I'm terrible at explaining and wiki says best ///
For example, if the authoritative name server for example.org is
ns1.example.org, a computer trying to resolve www.example.org first
resolves ns1.example.org. Since ns1 is contained in example.org, this
requires resolving example.org first, which presents a circular
dependency. To break the dependency, the name server for the top level
domain org includes glue along with the delegation for example.org.
The glue records are address records that provide IP addresses for
ns1.example.org. The resolver uses one or more of these IP addresses
to query one of the domain's authoritative servers, which allows it to
complete the DNS query.
Now you can disable the DNS cluster as you have just moved your DNS from Server A to Server B all while the actual nameservers haven't changed :)
When finished, I suggest using intodns.com to check all your results.