First of all, telnet is not secure. Use SSH instead, it works like telnet, but can be made secure. You can set SSH up with keys for authentication. See the tutorial section under networking on this board, there are how-to's there.
I have edited files on a remote system using vi or vim over ssh. You have to use root privlidges and set up SSH to allow root access. You also need the root password.
There are other ways. If you were going to do any amount of remote admin work on remote machines, install Webmin on the machines you want to administer. From your work station, use a web browser, and to connect, just
https://ip-address-of-remote:10000 . In Webmin you can set up users, and control a wide range of servers and system resources a user can control.
Hope this helps.