You must be missing some group definition in your example, since
\documentclass{article}
\begin{document}
Here is some text\footnote{First footnote}\par
Here is some text{\let\thefootnote\relax\footnotetext{Second footnote}}\par
Here is some text\footnote{Last footnote}
\end{document}
produces

The use of braces around the redefinition of \thefootnote - called scoping or grouping - makes ordinary modifications to macros only survive within the group. That is, their regular definitions return outside the group.
However, it's best to define some form of macro to take care of you particular \footnote construction. See Consistent typography. Here's an example that produces the same output:
\documentclass{article}
\setlength{\textheight}{10\baselineskip}% Just for this example
\makeatletter
\let\oldfootnote\footnote
\def\footnote{\@ifstar\footnote@star\footnote@nostar}
\def\footnote@star#1{{\let\thefootnote\relax\footnotetext{#1}}}
\def\footnote@nostar{\oldfootnote}
\makeatother
\begin{document}
Here is some text\footnote{First footnote}\par
Here is some text\footnote*{Second footnote}\par
Here is some text\footnote{Last footnote}
\end{document}