Na página de manual do crontab :
The entire command portion of the line, up to a newline or % character, will be executed by /bin/sh or by the shell specified in the SHELL variable of the crontab file. Percent-signs (%) in the command, unless escaped with backslash (\), will be changed into newline characters, and all data after the first % will be sent to the command as standard input.
Você deve escapar seus caracteres% em sua entrada crontab, da seguinte maneira:
* * * * * sudo tar -cvpzf /backupfolder/user1_backup-$(date +\%Y--\%m-\%d).tar.gz /home/students/user1