Using package 'libertine' overrides my sans-font with Libertine instead of Biolinum

The problem is \oldstylenums which switches the font. Just use the fontspec syntax.

\documentclass{book}

\usepackage{fontspec}
\usepackage{libertine}

%\setmainfont{Linux Libertine O}
%\setsansfont{Linux Biolinum O}

\usepackage{titlesec}

\titleformat{\chapter}[display]
  {\normalfont\sffamily\bfseries\LARGE}
  {\filright \sffamily\mdseries
    \fontsize{10em}{0em}\selectfont
    {\addfontfeature{Numbers=OldStyle}\thechapter}}
  {1em}
  {\filright}

\begin{document}
\chapter{Chapter 1}
\end{document}

The fontspec Way

The libertine package redefines \oldstylenums{} to switch the font. You can change back to a definition like the one from fontspec, which adds the OpenType font feature to the currently-selected font. Note that this is an incompatibility between the libertine and fontspec packages!

\documentclass{book}

\usepackage{fontspec}
\usepackage{libertine}

%% \oldstylenums and \liningums will change the style of the current font, as
%% in fontspec, not switch to the serif font, as in libertine.
\renewcommand\oldstylenums[1]{{\addfontfeatures{Numbers=OldStyle}{#1}}}
\renewcommand\liningnums[1]{{\addfontfeatures{Numbers=Lining}{#1}}}

\usepackage{titlesec}

\titleformat{\chapter}[display]
  {\normalfont\sffamily\bfseries\LARGE}
  {\filright \sffamily\mdseries
    \fontsize{10em}{0em}\selectfont
    \oldstylenums{\thechapter}}
  {1em}
  {\filright}

\begin{document}
\chapter{Chapter 1}

\oldstylenums{1234567890} vs. \liningnums{1234567890}.
\end{document}

You could alternatively give the command a new, unambiguous name or save the definitions after loading fontspec and before loading libertine.

The libertine Way

The libertine package provides a second command, \oldstylenumsf{}, to switch the font to Biolinum with old-style numbers. You could simply replace \textsf{\oldstylenums{}} with \oldstylenumsf{}.

The package defines the \biolinumOsF font:

\usepackage{libertine}

and

\titleformat{\chapter}[display]
{\normalfont\sffamily\bfseries\LARGE}
{\filright\mdseries\fontsize{10em}{0em}\selectfont
  \biolinumOsF\thechapter}
{1em}
{\filright}