sudo command > outputfile doesn't do what you think. Specifically, the outputfile is opened with your permissions, not the sudo user's (in this case, root). That's why it works when you become root and run the command - because then the redirect does run as root, who can write to /mnt/SVNBackup . It is necessary and sensible that sudo behaves this way.
Find some way of running the whole command as root; put it in root's cron, or write a one-line shell script and invoke that with sudo, or find another fix that works for you. In the limiting (and slightly stupid) case,
sudo svnadmin dump /usr/local/svn/repos/testrepo | sudo tee /mnt/SVNBackup/test1.svn
should produce the output both on stdout and in the file /mnt/SVNBackup/test1.svn . Bizarre though it may seem, if you really don't want the output, the simplest way may be to do
sudo svnadmin dump /usr/local/svn/repos/testrepo | sudo tee /mnt/SVNBackup/test1.svn > /dev/null
If the original svnadmin command produces stuff on STDERR and you want that in the log file too, try
sudo svnadmin dump /usr/local/svn/repos/testrepo 2>&1 | sudo tee /mnt/SVNBackup/test1.svn > /dev/null
Note that 2>&1 is a bash forumlation, if your shell is a csh-variant, or something different yet, you'll need to find something appropriate.
