Are there any vulnerabilities in having One server provide syslog and dns services?

From the zero information you give us the answer is "probably".

Try picking a Linux distribution and versions of the packages and research them.

All servers are vulnerable to something, new exploits are being discovered all the time.

You can mitigate and minimize attack footprints in all server implementations by uninstalling unused services and firewalling with ACLs anything that's left open.

...additionally,
did you perchance meant to say "risks" instead of "vulnerabilities"? Doug McIlroy back then explained Unix philosophy saying "write programs that do one thing and do it well" and this applies to service isolation as well. Usually one requires a syslog server for particular reasons and here confidentiality, integrity and availability have a different meaning (and result) compared to a public name server. Simply put using single purpose machines allows you to focus on securing and strengthening the server for its task.