Why does this naming of nodes via a foreach loop result in an error message even though it seems that the naming works partially?

The reason for the error message is already explained in CarLaTeX's comment.

I assume, you want to connect the nodes with lines. Then, the previous node can be remembered by option remember. The \if construct excludes the first loop, when there is only just one node defined yet.

\documentclass{article}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}

\foreach \index [remember=\index as \lastindex] in {0,1,2,3}{
  \node (v\index) at (\index cm, \index cm)[]{$v_{\index}$};
  \ifnum\index>0
    \draw (v\lastindex) -- (v\index);
  \fi
}

\end{tikzpicture}
\end{document}

A variant without \ifnum:

\documentclass{article}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}

\foreach \index in {0, 1, 2, 3} {
  \node (v\index) at (\index cm, \index cm)[]{$v_{\index}$};
}
\draw
  \foreach \index [remember=\index as \lastindex (initially 0)] in {1, 2, 3} {
    (v\lastindex) -- (v\index)
  }
;

\end{tikzpicture}
\end{document}

A possible solution is to define the first node out of the cicle:

\documentclass{amsart}   
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

    \begin{tikzpicture}

    \node (v0) {$v_0$};
    \foreach[evaluate=\index as \indexj using int(\index+1)] \index in {0,1,2}{
        \node (v\indexj) at (\indexj cm, \indexj cm)[]{$v_{\indexj}$};
        \draw (v\index)--(v\indexj);    
    }
    \end{tikzpicture}

\end{document}