line break after the word "proof" in proof environment

You just need to patch proof.

\documentclass{article}
\usepackage{amsmath,amsthm}

\usepackage{xpatch}

\xapptocmd{\proof}{\mbox{}\par\nobreak}{}{}

\textheight=6.2cm % just for the example

\begin{document}

\begin{proof}[Proof of my theorem]
    First line of my proof.

    Intermediary lines of my proof:
  \begin{itemize}
   \item next-to-last line
   \item last line of my proof
  \qedhere
  \end{itemize}
\end{proof}

\begin{proof}
    First line of my proof.

    Intermediary lines of my proof:
  \begin{itemize}
   \item next-to-last line
   \item last line of my proof
  \qedhere
  \end{itemize}
\end{proof}

\begin{proof}
This will be in the next page. Some filler text.
Some filler text. Some filler text. Some filler text.
Some filler text. Some filler text. Some filler text.
Some filler text. Some filler text. Some filler text.
Some filler text. Some filler text. Some filler text.
Some filler text. Some filler text. Some filler text.
Some filler text. Some filler text. Some filler text.
Some filler text. Some filler text. Some filler text.
Some filler text. Some filler text. Some filler text.
Some filler text. Some filler text. Some filler text.
\end{proof}

\end{document}

Just for testing, if you try and remove \nobreak, you'll see that the final “Proof” line will appear by itself at the bottom.

from your post, I understand that you need to set the heading Proof in a line and the following text to be in the next line, (if not so, please correct me). If I am correct, please try with the below code:

\documentclass{article}
\usepackage{amsmath,amsthm}

\makeatletter
\renewenvironment{proof}[1][\proofname]{\par
  \pushQED{\qed}%
  \normalfont \topsep6\p@\@plus6\p@\relax
  \trivlist
  \item[\hskip\labelsep
        \itshape
    #1\@addpunct{.}]~\newline
}{%
  \popQED\endtrivlist\@endpefalse
}
\makeatother

\begin{document}

\begin{proof}[Proof of my theorem]
    First line of my proof

    Intermediary lines of my proof:
  \begin{itemize}
   \item next-to-last line
   \item last line of my proof
  \qedhere
  \end{itemize}
\end{proof}

\begin{proof}
    First line of my proof

    Intermediary lines of my proof:
  \begin{itemize}
   \item next-to-last line
   \item last line of my proof
  \qedhere
  \end{itemize}
\end{proof}

\end{document}

You can apply Bruno Le Floch's magic powder (see below), but there is a simpler solution using the letltxmacro package, as egreg kindly pointed out. So, let's start with the simpler solution:

Redefining \proof using letltxmacro

\documentclass{article}
\usepackage{amsthm}
\usepackage{letltxmacro}
% \usepackage{lipsum} % uncomment to test the behavior at a page break

\makeatletter
\LetLtxMacro{\proof@ORIG}{\proof}

\renewcommand{\proof}[1][\proofname]{%
  \proof@ORIG[#1]\mbox{}\par\nobreak
}
\makeatother

\begin{document}

% To test the behavior at a page break, use \lipsum[1-4]\par\lipsum[5][1-11]
% here, then add a few words in order to see the proof pushed to the next page.
\begin{proof}[Proof of my theorem]
    First line of my proof.

    Intermediary lines of my proof:
  \begin{itemize}
   \item next-to-last line
   \item last line of my proof
  \qedhere
  \end{itemize}
\end{proof}

\begin{proof}
    First line of my proof.

    Intermediary lines of my proof:
  \begin{itemize}
   \item next-to-last line
   \item last line of my proof
  \qedhere
  \end{itemize}
\end{proof}

\end{document}

Without using letltxmacro

Note: this is not better than what precedes! The problem one faces with a naive redefinition of \proof using \let is due to the optional argument of \proof: \\proof (with two backslashes) is an intermediate macro defined by the \newcommand machinery internally used by \newenvironment; \\proof accepts \proof's optional argument as a delimited argument (TeX macros don't have the concept of optional arguments—LaTeX added this concept).

\\proof:
\long macro:[#1]->\par \pushQED (...)

Here is code that does what you want, based on Bruno Le Floch's magic powder:

\documentclass{article}
\usepackage{amsmath,amsthm}
% \usepackage{lipsum} % uncomment to test the behavior at a page break

% Redefinition that combines <https://tex.stackexchange.com/a/8091> (Bruno Le
% Floch) and <https://tex.stackexchange.com/a/85081> (Barbara Beeton).
% Actually, the latter has been modified a bit here after egreg's feedback.
% ;-)
\expandafter\let\expandafter\oldproof\csname\string\proof\endcsname
\let\oldendproof\endproof
\renewenvironment{proof}[1][\proofname]{%
  \oldproof[#1]\mbox{}\par\nobreak
}{%
  \oldendproof
}

\begin{document}

% To test the behavior at a page break, use \lipsum[1-4]\par\lipsum[5][1-11]
% here, then add a few words in order to see the proof pushed to the next page.
\begin{proof}[Proof of my theorem]
    First line of my proof.

    Intermediary lines of my proof:
  \begin{itemize}
   \item next-to-last line
   \item last line of my proof
  \qedhere
  \end{itemize}
\end{proof}

\begin{proof}
    First line of my proof.

    Intermediary lines of my proof:
  \begin{itemize}
   \item next-to-last line
   \item last line of my proof
  \qedhere
  \end{itemize}
\end{proof}

\end{document}