Yes, the permissions are going to be reset every time you plug it in, because /dev/ttyUSB0 isn't a traditional file. It's created dynamically by the device manager (udev) when it detects new hardware being plugged in.
The simplest way around that is to just run CHIRP with root privs, i.e. sudo chirpw.
Alternatively if you didn't want to do that, you could write a udev rule that will set the permissions automatically for you. To do that, you'll have to copy /lib/udev/rules.d/50-udev-default.rules to /etc/udev/rules.d/50-udev-default.rules and then edit the new file. Look for the comment that says serial, that's where the rules for serial devices are. Below that, you could add a rule that sets the permissions of /dev/ttyUSB0 to 666. Mine now looks like this:
# serial
KERNEL=="tty[A-Z]*[0-9]|pppox[0-9]*|ircomm[0-9]*|noz[0-9]*|rfcomm[0-9]*", GROUP="dialout"
# modify permissions just for ttyUSB0
KERNEL=="ttyUSB0", MODE="0666"
Now when you plug in the cable, it will automatically set the permissions for you and you won't have to run the software as root.