Renewcommand labelitem doesn't work with multiple languages

That is french mocking about, try adding this

\addto\captionsenglish{
  \renewcommand{\labelitemi}{$\bullet$}
}

Basically french alters lists, but english does not (AFAIR). So we add such a switch back to english

---

Edit, to ignore french all together, use

\usepackage[shortlabels]{enumitem}
\setlist[itemize]{label=\textbullet}

enumitem is a very good package to know, the shortlabels option makes it a 1-1 replacement for the enumerate package.

Use \frenchbsetup{StandardItemLabels=true}, so babel-french will not use French style itemized lists.

\documentclass[10pt,a4paper]{report}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[english,french]{babel}
\usepackage{amsmath,amsthm}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{enumerate}

\frenchbsetup{StandardItemLabels=true}

\begin{document}

\selectlanguage{english}

\chapter{First chapter}

This is my list:

\begin{itemize}
\item Value 1
\item Value 2
\end{itemize}

\selectlanguage{french}

\chapter{Second chapitre}

Ceci est ma liste:

\begin{itemize}
\item Valeur 1
\item Valeur 2
\end{itemize}

\end{document}

(I used a devious trick to get the output in a single page, not shown here.) Note that with French it is highly recommended to have \usepackage[T1]{fontenc}. I can't recommend utf8x.

You can use the iflang package, together with enumitem, instead of enumerate. Note that maybe you'll also have to specify spacings as frenchbhands over to enumitem. If you want to come back with the bullet symbol, even in the French part, just use \setlistitemize,1]{label=\textbullet}. Btw, I think the itemize marker in french should be an endash.

\documentclass[10pt, a4paper]{report}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[english,french]{babel}
\usepackage{iflang}

\usepackage{enumitem}

\setlist[itemize, 1]{label = \IfLanguageName{french}{\textendash}{\textbullet}}

\begin{document}

\selectlanguage{english}

\chapter{First chapter}

This is my list :

 \begin{itemize}
    \item Value 1
    \item Value 2
 \end{itemize}

 \selectlanguage{french}
 \chapter{Second chapitre}

 Ceci est ma liste:

 \begin{itemize}
    \item Valeur 1
    \item Valeur 2
 \end{itemize}


 \chapter{Chapitre troisième}

\setlist[itemize, 1]{label = \textbullet}
 Ceci est ma liste:

 \begin{itemize}
    \item Valeur 1
    \item Valeur 2
 \end{itemize}

 \end{document}