The dot seems to be defined in the macro \@thm with the line
\thm@headpunct{.}% add period after heading
I use xpatch to change that.
\documentclass{article}
\usepackage{amsthm,xpatch}
\makeatletter
\xpatchcmd{\@thm}{.}{}{}{}
\makeatother
\begin{document}
\makeatletter
\newtheorem*{definition*}{Definition}
\begin{definition*}
Here is the definition
\end{definition*}
\end{document}

For reference, here is the definition of \@thm in amsthm.sty:
\def\@thm#1#2#3{%
\ifhmode\unskip\unskip\par\fi
\normalfont
\trivlist
\let\thmheadnl\relax
\let\thm@swap\@gobble
\thm@notefont{\fontseries\mddefault\upshape}%
\thm@headpunct{.}% add period after heading
\thm@headsep 5\p@ plus\p@ minus\p@\relax
\thm@space@setup
#1% style overrides
\@topsep \thm@preskip % used by thm head
\@topsepadd \thm@postskip % used by \@endparenv
\def\@tempa{#2}\ifx\@empty\@tempa
\def\@tempa{\@oparg{\@begintheorem{#3}{}}[]}%
\else
\refstepcounter{#2}%
\def\@tempa{\@oparg{\@begintheorem{#3}{\csname the#2\endcsname}}[]}%
\fi
\@tempa
}
amsthmor thentheorempackage. – Mico Feb 05 '16 at 19:19amsthm? – Werner Feb 05 '16 at 19:34