You can't delete them because you don't have permission. To get permission, you'll have to convince the owner of the server to give you permission. This doesn't necessary mean you'll get root permission, though it might.
Curiously, I am facing the exact same problem. I have inherited a server that is not being maintained and the log files are many gigabytes large. Right now I am going back and forth with my client and the owner of the server (not the same person). Until someone does something (such as give me root access) my client's work is at a standstill because I need to access those logs and their huge size is making that impossible via a web interface. The server owner is (1) being lazy about maintaining the server and (2) dragging his heels about giving me root access.