Many people do not recommend to remove Asus Live Update from your PC to ensure that you remain updated with the latest BIOS version and the device drivers.
Although it is unlikely that Asus Live Update would prevent you from browsing the Internet (unless your entire Internet bandwidth is being consumed by the program to download the new drivers), if you want to remove the tool, you can do so as it would not harm your system.
However before you start removing Asus Live Update, make sure to create a restore point on your Windows. You can get the detailed instructions on creating restore points on your Windows here
To add another layer of protection, create a backup image of your system drive using any ghosting program. Instructions on how you can do this using one such software named Acronis TrueImage can be found here
Even though it is unlikely that your system becomes unstable after removing Asus Live Update, if it happens, you can restore the system to the restore point that you created.
In the worst case scenario, if your computer fails to boot altogether or starts functioning improperly, you can restore Windows using the backup image that you created the using ghosting program.
Hope this is of some help.