Unexplained Overfull Box

There are two things that cause this:

1. Paragraph indentation. The standard paragraph indentation is added before the tikzpicture, you can disable that for a single line with noindent, i.e. noindenttikz...

2. When you draw a line, the bounding box will actually extend by half the line width beyond the end coordinates, as you can see from this example:

    documentclass[border=5mm]{standalone}
    usepackage{tikz}
    begin{document}
   
    begin{tikzpicture}
      draw [line width=5mm] (0,0) -- (1,0);
      draw [very thin, red] (0,-0.5) -- (0,0.5);
   
      draw [thin, blue] (current bounding box.south east) rectangle (current bounding box.north west);
    end{tikzpicture}
    end{document}
   

The blue line indicates the bounding box.

If you add line cap=rect, the line will extend to fill that blank space, and you pgf has the handy macro pgflinewidth that has the width of the current path. Hence, you can do draw[thick,line cap=rect] (0,0) -- (linewidth-pgflinewidth,0) ....

Working example, as egreg mentioned you shouldn't use minimal (Why should the minimal class be avoided?):

documentclass{article}

usepackage{tikz}

newcommand{TotalMarks}[1]{%
    tikzdraw[thick,line cap=rect]
    (0,0) -- (linewidth-pgflinewidth,0) -- 
    ++(0,2em) -- ++(-2em,0) -- 
    ++(0,-2em) node[pos=0.6,left] 
        {makebox[2.9cm]{textbf{Total: #1 marks}hfill}};}

begin{document}
noindentTotalMarks{2}
end{document}