Footnote in align environment

amstext.sty defines a \iffirstchoice@ test just for this: it is true in the first branch of the inner \mathchoice called by \text and false in the remaining three.

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\newcommand{\AlignFootnote}[1]{%
  \ifmeasuring@
  \else
    \iffirstchoice@
      \footnote{#1}%
    \fi
  \fi}
\makeatother

\begin{document}
\begin{minipage}{0.5\linewidth}
\begin{align*}
    E &= mc^3 &&\text{Einstein}\AlignFootnote{Better check this.} \\ 
    F &= mb   &&\text{Newton\AlignFootnote{This also seems wrong for some reason.}}
\end{align*}
\end{minipage}
\end{document}

Actually the size of the footnote mark is not taken into consideration for the size of the equation, so a more correct answer would be

\makeatletter
\newcommand{\AlignFootnote}[1]{%
  \ifmeasuring@
    \chardef\@tempfn=\value{footnote}%
    \footnotemark
    \setcounter{footnote}{\@tempfn}%
  \else
    \iffirstchoice@
      \footnote{#1}%
    \fi
  \fi}
\makeatother

that prints only the footnotemark and restores the counter at its previous value.

Limitation If you say \dfrac{\text{Newton\AlignFootnote{X}}}{2} you won't get the footnote. But this shouldn't be a big deal.

If you want to use the \footnote command, this works

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

\makeatletter
\let\original@footnote\footnote
\newcommand{\align@footnote}[1]{%
  \ifmeasuring@
    \chardef\@tempfn=\value{footnote}%
    \footnotemark
    \setcounter{footnote}{\@tempfn}%
  \else
    \iffirstchoice@
      \original@footnote{#1}%
    \fi
  \fi}
\pretocmd{\start@align}{\let\footnote\align@footnote}{}{}
\makeatother

\begin{document}
\begin{minipage}{0.5\linewidth}
\begin{align*}
    E &= mc^3 &&\text{Einstein\footnote{Better check this.}} \\
    F &= mb   &&\text{Newton\footnote{This also seems wrong for some reason.}}
\end{align*}
\end{minipage}
\end{document}

Note. The above codes only work for when we're inside minipage. For outer footnotes the only way is to use \footnotemark and \footnotetext, with some adjustments not to make the counter advance too many times.

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

\makeatletter
\let\original@footnotemark\footnotemark
\newcommand{\align@footnotemark}{%
  \ifmeasuring@
    \chardef\@tempfn=\value{footnote}%
    \original@footnotemark
    \setcounter{footnote}{\@tempfn}%
  \else
    \iffirstchoice@
      \original@footnotemark
    \fi
  \fi}
\pretocmd{\start@align}{\let\footnotemark\align@footnotemark}{}{}
\makeatother

\textheight=4cm % just to produce a smaller picture

\begin{document}
\begin{align*}
    E &= mc^3 &&\text{Einstein\footnotemark} \\
    F &= mb   &&\text{Newton\footnotemark}
\end{align*}
\addtocounter{footnote}{-1}
\footnotetext{Better check this.} 
\addtocounter{footnote}{1}
\footnotetext{This also seems wrong for some reason.}
\end{document}

Code

\documentclass{article}
\usepackage{amsmath}
\usepackage{letltxmacro}
\makeatletter
\newcommand*{\fakeFN}[1]{\@makefnmark}
\newcommand{\AlignFootnote}[1]{%
    \ifmeasuring@% I want to use \@makefnmark here, too,
                 % but \@thefnmark is undefined.
                 % The substitute "kerning" (see \else branch) is a work-around, but
                 % messes with align's spacing.
    \else%
        \footnote{#1}\sbox0{\@makefnmark}\kern-\wd0%
    \fi
}
\makeatother

\newdimen\myparboxwidth
\newcommand{\textfn}[1]{% I need to increment the fnmark counter here …
  {\LetLtxMacro\footnote\fakeFN\sbox0{#1}\global\myparboxwidth=\wd0}
  \parbox{\myparboxwidth}{#1}%
}

%\usepackage{lua-visual-debug} % for the second output
\begin{document}
\noindent\begin{minipage}{\linewidth} 
\begin{align*} 
E &= mc^3 &&\text{Einstein}\AlignFootnote{Better check this.} \\
F &= mb &&\textfn{Newton\AlignFootnote{This also seems wrong for some reason.}}
\end{align*}
\end{minipage}

\begin{align*}
E &= mc^2 &&\text{Einstein is happy now.} \\ 
F &= ma &&\text{Newton, too.}
\end{align*} 
\end{document}

Output

Output with lua-visual-debug activated