Ive read that its okay - but obviously not pickled! and of course you would need to be very careful of the staining (unless you are going for a purple hamster look
) I think the book I read it in actually said to feed raw rather than cooked too - I'll have a look.
You can get beetroot treats - mine don't go crazy over them however.
Edit: the Roy Robinson/David Baglin book says it is okay, cut into small cubes.