I have taken the code from Is it possible to distinguish between the soft copy and the hard copy of a pdf? and reversed the view/print states to define a viewonly environment, then wrapped \BR@backref in that environment. It appears to work but note that the content is only invisible when printed, but still occupies space, and so may result in blank lines in printed editions.
Note that it is not necessary to actually print the document to see if it works; the print preview mode should show the difference.
Update: Now uses ocg-p package version 0.4 (or greater) for more concise and reliable code. (Old answer available through revision history.)
Example here uses a test .bib file that should come with your distribution.
\documentclass[10pt,a4paper]{article}
\usepackage[a4paper]{geometry}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{natbib}
\bibliographystyle{plainnat}
\usepackage[backref=page]{hyperref}
% "viewonly" environment for content which should only be visible when the document is viewed on screen, not when it is printed
\usepackage{ocg-p}
\newenvironment{viewonly}{\begin{ocg}[printocg=never,listintoolbar=never]{Soft copy only}{viewonly}{1}}{\end{ocg}}
\makeatletter
\let\oldBR@backref=\BR@backref
\renewcommand{\BR@backref}[1]{\begin{viewonly}\oldBR@backref{#1}\end{viewonly}}
\makeatother
\begin{document}
This text is always visible. \cite{GSM97,RudolphGrabowskiGraubmann96}
\begin{viewonly}
This can not be seen if the document is printed.
\end{viewonly}
This text is visible in both the soft and the hard copy. \cite{Lam94}
\bibliography{biblio}
\end{document}
