Extra spacing around restatable theorems

The problem with your solution is that restatable only introduces extra space when it is preceded by another environment. In the other cases, spacing is correct, and the \vspace should be avoided.

By doing several experiments, it seems the culprit is the \label{thmt@@#1@data} in \thmt@rst@storecounters. By removing the \label command, spacing is correct, but something else probably will go wrong. My solution, which I am testing in these days, is to move the call to \thmt@rst@storecounters in \thmt@restatable. This is the version of \thmt@restatable I am testing right now:

\renewenvironment{thmt@restatable}[3][]{%
  \thmt@toks{}% will hold body
  \stepcounter{thmt@dummyctr}% used for data storage label.
  \long\def\thmrst@store##1{%
    \@xa\gdef\csname #3\endcsname{%
      \@ifstar{%
        \thmt@thisistheonefalse\csname thmt@stored@#3\endcsname
      }{%
        \thmt@thisistheonetrue\csname thmt@stored@#3\endcsname
      }%
    }%
    \@xa\long\@xa\gdef\csname thmt@stored@#3\@xa\endcsname\@xa{%
      \begingroup
      \ifthmt@thisistheone
        % nothing here in my patched version
      \else
        % this one should use other numbers...
        % first, fake the theorem number.
        \@xa\protected@edef\csname the#2\endcsname{%
          \thmt@trivialref{thmt@@#3}{??}}%
        % if the number wasn't there, have a "re-run to get labels right"
        % warning.
        \ifcsname r@thmt@@#3\endcsname\else
          \G@refundefinedtrue
        \fi
        % prevent stepcountering the theorem number,
        % but still, have some number for hyperref, just in case.
        \@xa\let\csname c@#2\endcsname=\c@thmt@dummyctr
        \@xa\let\csname theH#2\endcsname=\theHthmt@dummyctr
        % disable labeling.
        \let\label=\@gobble
        \let\ltx@label=\@gobble% amsmath needs this
        % We shall need to restore the counters at the end
        % of the environment, so we get
        % (4.2) [(3.1 from restate)] (4.3)
        \def\thmt@restorecounters{}%
        \@for\thmt@ctr:=\thmt@innercounters\do{%
          \protected@edef\thmt@restorecounters{%
            \thmt@restorecounters
            \protect\setcounter{\thmt@ctr}{\arabic{\thmt@ctr}}%
          }%
        }%
        % pull the new semi-static definition of \theequation et al.
        % from the aux file.
        \thmt@trivialref{thmt@@#3@data}{}%
      \fi
      % call the proper begin-env code, possibly with optional argument
      % (omit if stored via key-val)
      \ifthmt@restatethis
        \thmt@restatethisfalse
      \else
        \csname #2\@xa\endcsname\ifx\@nx#1\@nx\else[{#1}]\fi
      \fi
      \ifthmt@thisistheone
         % these are the valid numbers, store them for the other
         % occasions.
         \thmt@rst@storecounters{#3}%
        % store a label so we can pick up the number later.
        \label{thmt@@#3}%
      \fi
      % this will be the collected body.
      ##1%
      \csname end#2\endcsname
      % if we faked the counter values, restore originals now.
      \ifthmt@thisistheone\else\thmt@restorecounters\fi
      \endgroup
    }% thmt@stored@#3
    % in either case, now call the just-created macro,
    \csname #3\@xa\endcsname\ifthmt@thisistheone\else*\fi
    % and artificially close the current environment.
    \@xa\end\@xa{\@currenvir}
  }% thm@rst@store
  \thmt@collect@body\thmrst@store
}{%
  %% now empty, just used as a marker.
}

I added one line to the definition of the \thmt@rst@storecounters macro: a hardwired \vspace. It might not be pretty, but it gets the job done. (EDITED to make the fix to a smaller routine, so as to take up less space. The ORIGINAL post had the same fix applied to the much longer thmt@restatable environment.)

\documentclass{article}

\usepackage{amsthm}
\usepackage{thmtools}

\makeatletter

\def\thmt@rst@storecounters#1{%
%THIS IS THE LINE I ADDED:
\vspace{-1.9ex}%
  \bgroup
        % ugly hack: save chapter,..subsection numbers
        % for equation numbers.
  %\refstepcounter{thmt@dummyctr}% why is this here?
  %% temporarily disabled, broke autorefname.
  \def\@currentlabel{}%
  \@for\thmt@ctr:=\thmt@innercounters\do{%
    \thmt@sanitizethe{\thmt@ctr}%
    \protected@edef\@currentlabel{%
      \@currentlabel
      \protect\def\@xa\protect\csname the\thmt@ctr\endcsname{%
        \csname the\thmt@ctr\endcsname}%
      \ifcsname theH\thmt@ctr\endcsname
        \protect\def\@xa\protect\csname theH\thmt@ctr\endcsname{%
          (restate \protect\theHthmt@dummyctr)\csname theH\thmt@ctr\endcsname}%
      \fi
      \protect\setcounter{\thmt@ctr}{\number\csname c@\thmt@ctr\endcsname}%
    }%
  }%
  \label{thmt@@#1@data}%
  \egroup
}%

\makeatother

\declaretheorem{theorem}

\begin{document}

\begin{restatable*}{theorem}{mythm}
This is a restated theorem.
\end{restatable*}

\begin{theorem}
This is a regular theorem.
\end{theorem}
\begin{theorem}
This is a regular theorem.
\end{theorem}

\mythm
\mythm
\mythm

\begin{theorem}
This is a regular theorem.
\end{theorem}
\begin{theorem}
This is a regular theorem.
\end{theorem}
\begin{theorem}
This is a regular theorem.
\end{theorem}

\mythm*
\mythm*
\mythm*

\end{document}