You could a) disable \clearpage inside a group b) add a \chapter*{Bibliography} and patch \thebibliography so that it uses \section* instead of \chapter*.
EDIT: Added code to correct running headers for \section* (sub)bibliographies.
\documentclass{book}
\usepackage{etoolbox}
\makeatletter
\newcommand*{\patchthebibliography}{%
\patchcmd{\thebibliography}{\chapter*}{\section*}{}{}%
\patchcmd{\thebibliography}{%
\@mkboth{\MakeUppercase\bibname}{\MakeUppercase\bibname}%
}{%
\markright{\MakeUppercase\bibname}%
}{}{} %
}
\makeatother
\newcounter{firstbib}
\begin{document}
\chapter{foo}
Some text \cite{A01,B02}.
\renewcommand{\bibname}{Literary works}
\begin{thebibliography}{9}
\bibitem{A01} A. Author. \emph{Alpha}. 2001.
\setcounter{firstbib}{\value{enumiv}}
\end{thebibliography}
\begingroup
\let\clearpage\relax
\renewcommand{\bibname}{Web references}
\begin{thebibliography}{9}
\setcounter{enumiv}{\value{firstbib}}
\bibitem{B02} B. Buthor. \emph{Bravo}. 2002.
\end{thebibliography}
\endgroup
\begingroup
\patchthebibliography
\chapter*{Bibliography}
\markboth{\MakeUppercase{Bibliography}}{\MakeUppercase{Bibliography}}
\renewcommand{\bibname}{Literary works}
\begin{thebibliography}{9}
\bibitem{A01} A. Author. \emph{Alpha}. 2001.
\setcounter{firstbib}{\value{enumiv}}
\end{thebibliography}
\renewcommand{\bibname}{Web references}
\begin{thebibliography}{9}
\setcounter{enumiv}{\value{firstbib}}
\bibitem{B02} B. Buthor. \emph{Bravo}. 2002.
\end{thebibliography}
\end{document}
