biblatex
Here is a solution for biblatex. We can (ab)use the mark mechanism, each entry sets the marks and then the header grabs \rightmark and \leftmark, which contain the first and last reference number on each page.
\documentclass[british]{book}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[style=numeric, backend=biber]{biblatex}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead[LE]{\slshape My text}
\fancyhead[RE]{}
\fancyhead[LO]{\hfill \slshape \leftmark}
\defbibheading{bibliography}[\bibname]{%
\chapter*{#1}%
\fancyhead{}%
\fancyhead[RO,LE]{#1 \rightmark{} -- \leftmark}}
\makeatletter
\providecommand*{\@mkdouble}[1]{\@mkboth{#1}{#1}}
\newcommand*{\markbib}{%
\@mkdouble{\thefield{labelprefix}\thefield{labelnumber}}}
\makeatother
\renewbibmacro*{begentry}{\markbib}
\addbibresource{biblatex-examples.bib}
\begin{document}
\nocite{*}
\printbibliography
\end{document}

thebibliography/BibTeX
A structurally similar approach works for standard thebibliography (the definitions may need to be adjusted if you use a different document class or a more sophisticated bibliography package like natbib).
Things get a bit more @y because we have to redefine the complete thebibliography environment in order to change the headers and \@lbibitem as well as \@bibitem to get the citation label into the mark mechanism.
\documentclass[british]{book}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead[LE]{\slshape My text}
\fancyhead[RE]{}
\fancyhead[LO]{\hfill \slshape \leftmark}
\makeatletter
% \@lbibitem & \@bibitem from latex.ltx
% the only change apart from indentation is the addition of \markbib
\def\@lbibitem[#1]#2{%
\item[\@biblabel{#1}\hfill]%
\markbib{#1}% <-- added
\if@filesw
{\let\protect\noexpand
\immediate\write\@auxout
{\string\bibcite{#2}{#1}}}%
\fi
\ignorespaces}
\def\@bibitem#1{%
\item
\markbib{\the\value{\@listctr}}% <-- added
\if@filesw
\immediate\write\@auxout
{\string\bibcite{#1}{\the\value{\@listctr}}}%
\fi
\ignorespaces}
% from book.cls
% only change was to add the two \fancyhead calls
\renewenvironment{thebibliography}[1]
{\chapter*{\bibname}%
\fancyhead{}% <-- added
\fancyhead[RO,LE]{\bibname{} \rightmark{} -- \leftmark}% <-- added
\list{\@biblabel{\@arabic\c@enumiv}}%
{\settowidth\labelwidth{\@biblabel{#1}}%
\leftmargin\labelwidth
\advance\leftmargin\labelsep
\@openbib@code
\usecounter{enumiv}%
\let\p@enumiv\@empty
\renewcommand\theenumiv{\@arabic\c@enumiv}}%
\sloppy
\clubpenalty4000
\@clubpenalty \clubpenalty
\widowpenalty4000%
\sfcode`\.\@m}
{\def\@noitemerr
{\@latex@warning{Empty `thebibliography' environment}}%
\endlist}
\providecommand*{\@mkdouble}[1]{\@mkboth{#1}{#1}}
\newcommand*{\markbib}{\@mkdouble}
\makeatother
\begin{document}
\nocite{*}
\begin{thebibliography}{99}
\bibitem{example01}
\bibitem{example02}
\bibitem{example03}
\bibitem{example04}
\bibitem{example05}
\bibitem{example06}
\bibitem{example07}
\bibitem{example08}
\bibitem{example09}
\bibitem{example10}
\bibitem{example11}
\bibitem{example12}
\bibitem{example13}
\bibitem{example14}
\bibitem{example15}
\bibitem{example16}
\bibitem{example17}
\bibitem{example18}
\bibitem{example19}
\bibitem{example20}
\bibitem{example21}
\bibitem{example22}
\bibitem{example23}
\bibitem{example24}
\bibitem{example25}
\bibitem{example26}
\bibitem{example27}
\bibitem{example28}
\bibitem{example29}
\bibitem{example30}
\bibitem{example31}
\bibitem{example32}
\bibitem{example33}
\bibitem{example34}
\bibitem{example35}
\bibitem{example36}
\bibitem{example37}
\bibitem{example38}
\bibitem{example39}
\bibitem{example40}
\bibitem{example41}
\bibitem{example42}
\bibitem{example43}
\bibitem{example44}
\bibitem{example45}
\bibitem{example46}
\bibitem{example47}
\end{thebibliography}
\end{document}