My father's LG G3 has stopped working. after logo screen it just shuts down.
Is there a way to boot the phone as only a hard drive from a PC ?
He has important notes on the phone that he needs to restore.
No, because you have to boot up your phone to use the MTP functions like HDD (or External Mass Storage). Probably your father did a NAND erase, which delete the boot-up partition. You could use LG Flash Tool to install the .kdz and make the new partition.