Is it possible to use fancyvrb, eso-pic and TikZ simultaneously?

The verbatim settings are not being reset in the page head overlay from eso-pic as a workaround you could do

\documentclass{article}
\usepackage{fancyvrb}
\usepackage{tikz}
\usepackage{eso-pic}

\AddToShipoutPictureFG{%
    \catcode`$=3 
    \begin{tikzpicture}[remember picture, overlay]
        \node[anchor=north east] at (current page.north east) {Hello};
    \end{tikzpicture}}
\begin{document}

\tikz{\fill[blue](0,0)rectangle(1,1);}
\VerbatimInput{tikzducks.sty}
\end{document}

The error message is here a false positive: Even if the picture would use $, as they are already tokenized they would work fine. The check from tikz doesn't prevent following errors as it doesn't reset the catcode. I would redefine the tikz command that checks the catcode of the $ either to relax or to a warning. Imho this is quite enough.

\documentclass{article}
\usepackage{fancyvrb}
\usepackage{tikz}

\makeatletter
% or \let\tikz@ensure@dollar@catcode\relax
\def\tikz@ensure@dollar@catcode{%
  \ifnum\catcode`\$=3 %
  \else
    \pgfutil@packagewarning{tikz}{Sorry, some package has redefined the meaning of the
      math-mode dollar sign. This is incompatible with tikz and its calc
      library and might cause unrecoverable errors}%
    % only show warning once:
    \global\let\tikz@ensure@dollar@catcode=\relax
  \fi
}%

\makeatother
\usepackage{eso-pic}
\AddToShipoutPictureFG{%
    \begin{tikzpicture}[remember picture, overlay]
        \node[anchor=north east] at (current page.north east) {Hello};
    \end{tikzpicture}}
\begin{document}
\tikz{\fill[blue](0,0)rectangle(1,1);}
\VerbatimInput{tikzducks.sty}
\end{document}