It's been many years since I used rsh, rdump, or any of the other r stuff. It is insecure and deprecated, especially for root. I believe the .rhosts file has to be in root's home directory if you want root to be able to login. Check the man page. That is dangerous, at least in part because it is unencrypted and because it would allow a root compromise on one machine to easily spread to another machine.
What I have done to accomplish this securely is to create a backup user account and set it up with public key authentication for ssh. See
http://sial.org/howto/openssh/publickey-auth/ for an excellent howto. Then, I can run the dump (or tar or whatever) as root and pipe it through the backup user to an ssh into the remote system and onto the tape drive. It may seem a bit contorted or complex, but you just have to try each step and get it working.
An example with ufsdump on Solaris would be:
ufsdump 0cnlTfuN 2h - ${RDEV} ${WHICHSNAP} | su - backup -c "ssh ${TAPESERVER} \"dd obs=64b of=${TAPDEV}\"";
(That's all on one line.)
So, the backup data gets streamed through the ssh pipe and picked up by dd which puts it onto the tape.
