Disabling drush backups after drush up?
Yes, use the --no-backup option with your command. 

drush --no-backup up myproject


If you're you're looking to make this a default option, you can do something like Bash aliasing and just run that alias instead:

alias drush-no-bup='drush --no-backup'


Then run: 

drush-no-bup up myproject