The tabular environment "gets confused" by the & characters of flalign* environment, so you need to hide them using, for example, a pair of braces (or a box, like you found out with \fbox):
\documentclass[11pt]{amsart}
\begin{document}
\noindent\begin{minipage}{1\textwidth}
\begin{tabular}{@{}l@{}r@{}}
\begin{minipage}[t]{.65\textwidth}
\begin{verbatim}
\begin{flalign*}
P &=x^2+y^+2xy &\\
&=(x+y)^2
\end{flalign*}
\end{verbatim}
\end{minipage}%
&
{%
\begin{minipage}[t]{.35\textwidth}
\begin{flalign*}
P &=x^2+y^2+2xy &\\
&=(x+y)^2
\end{flalign*}
\end{minipage}%
}
\end{tabular}
\end{minipage}
\end{document}
I did some modifications (removed spurious blank spaces, removed inter-column space) to your code to prevent overfull boxes. Do you really need that somehow complicated nesting structure? At least in this example you can get the same result just with
\documentclass[11pt]{amsart}
\begin{document}
\noindent\begin{minipage}{\textwidth}
\begin{minipage}[t]{.65\textwidth}
\begin{verbatim}
\begin{flalign*}
P &=x^2+y^+2xy &\\
&=(x+y)^2
\end{flalign*}
\end{verbatim}
\end{minipage}%
\begin{minipage}[t]{.35\textwidth}
\begin{flalign*}
P &=x^2+y^2+2xy &\\
&=(x+y)^2
\end{flalign*}
\end{minipage}
\end{minipage}
\end{document}
If you're interested in showing code and output side-by-side, perhaps the showexpl package could be of interest to you.
It's not clear to me why you have minipage environments encasing tabular environments. Here's a solution that dispenses with the minipages and top-aligns two tabular environments within a tabularx environment.
To avoid overlaps between the \hlines drawn by the outer tabularx and the inner tabular environments, I've omitted some of your \hline instructions from the inner environments. If you decide to get rid of the outer \hlines, be sure to re-insert the \hlines in the inner tabulars.
\documentclass{article}
\usepackage{tabularx}
\begin{document}
\noindent
\begin{tabularx}{\textwidth}{@{}X@{}X@{}}
\hline
\begin{tabular}[t]{|l|l|}
1 & 2 \\ \hline
3 & 4 \\ \hline
5 & 6
\end{tabular}
&
\begin{tabular}[t]{|l|l|}
7 & 8 \\ \hline
\end{tabular}\\ \hline
\end{tabularx}
\end{document}
Next, here's a version with three vertical bars drawn by the outer tabularx (far left, middle, and right). To avoid overlap of the vertical lines drawn by the inner and outer table-like environments, it may again be a good idea to omit the first vertical bar from each of the inner tabulars.
\documentclass{article}
\usepackage{tabularx}
\begin{document}
\noindent
\begin{tabularx}{\textwidth}{|@{}X|@{}X@{}|}
\hline
\begin{tabular}[t]{l|l|}
1 & 2 \\ \hline
3 & 4 \\ \hline
5 & 6
\end{tabular}
&
\begin{tabular}[t]{l|l|}
7 & 8 \\ \hline
\end{tabular}\\ \hline
\end{tabularx}
\end{document}
Finally, for the sake of completeness, here's the code that's needed if you wish to use minipage environments after all to typeset the preceding example. Note that it's necessary to use the [t] position specifier for both minipages and both tabulars in order to get the vertical alignment just right. (Not showing a separate screenshot since the result is identical to the one above.)
\documentclass{article}
\usepackage{tabularx}
\begin{document}
\noindent
\begin{tabularx}{\textwidth}{|@{}X|@{}X@{}|}
\hline
\begin{minipage}[t]{\linewidth}
\begin{tabular}[t]{l|l|}
1 & 2 \\ \hline
3 & 4 \\ \hline
5 & 6
\end{tabular}
\end{minipage}
&
\begin{minipage}[t]{\linewidth}
\begin{tabular}[t]{l|l|}
7 & 8 \\ \hline
\end{tabular}
\end{minipage}\\
\hline
\end{tabularx}
\end{document}
Best Answer
This probably achieves what you want (I made the code compilable by replacing
\fby\fracand changed the minipage alignment to[h]):Just for the records: if I wanted to get the vertical line, I'd use
Note that I kicked out some of the unnecessary packages and also removed one of the conflicting options (letterpaper).