If you do not require root privileges at all to start the server, you can use something like
sudo -u <username> ./opt/teamspeak3-server_linux-x86/ts3server_startscript.sh start
to start it as user <username>.
This command line must be placed in a file that is executed during startup.
IIRC there is a file for local stuff named somewhat like 99-local in /etc/init.d/. If not, you could write one yourself using some other startup file as template. To schedule the file for being run during startup, use the following command:
chkconfig --level 345 <your file name> on
whereas <your file name> is relative to /etc/init.d/.
Please note that you should specify sudo with its full path (eg. /usr/bin/sudo) to prevent the possibility of executing another version of sudo placed somewhere in the search path and you should resolve the . in ./opt/... (if executed from the root directory, it would be /opt/...).