mysqldump will need a password for the mysql user root. If you don't supply that password it won't work,
sudo or no
mysqldump can look at your
.my.cnf file to get the password, and it'll get that from the user runing the
mysqldump command. So, if the Linux root user has a .my.cnf configured with a password in it, then yes,
sudo will help. However, if the mysql root user has a password and it's not configured in any
sudo won't help.
You could add the mysql root user and password to your own user's
.my.cnf and then you don't need
sudo either, but that's a security risk.
As the other answers point out as well, you might need
sudo to write the output somewhere that root (Linux, not mysql) owns.
Lastly, you don't tell us which user is executing the script, if it's in root's crontab (for example), then
sudo will not be necessary, although you may still need a
.my.cnf to provide the password.
If your mysql root user has no password set, then the only issue is writing the output, in which case
sudo is required if you don't run the script as root.
If you run the script as root (for example, as a script in
/etc/cron.daily) then you should not use sudo within the script, irrespective of all the above comments.