A line of text with proper positioning but consuming no space?

You can locally disable the interline skip mechanism:

\documentclass{article}
\usepackage{tabu}

\begin{document}

\noindent
\begin{tabu}{|X[m,c]|X[-2,m,c]|X[m,c]|} \tabucline
    \scriptsize This is something \linebreak on the left
    &
    \offinterlineskip
    \Huge\strut Here's a Title\par
    \vspace{-2pt}
    \normalsize\strut And a subtitle
    &
    \scriptsize And here's some stuff \linebreak on the \linebreak right side
    \\\tabucline\\
\end{tabu}

\end{document}

The \vspace{-2pt} seems necessary, particularly because the huge title has no descenders. By inserting struts you ensure that the characters don't touch the lines.

Alternatively, put the subtitle in another row:

\noindent
\begin{tabu}{|X[m,c]|X[-2,m,c]|X[m,c]|} \tabucline
    \scriptsize This is something \linebreak on the left
    &
    \Huge\strut Here's a Title
    &
    \scriptsize And here's some stuff \linebreak on the \linebreak right side
    \\
    & And a subtitle & \\
\tabucline\\
\end{tabu}

Based on the comments you seem to want to make sure that the subtitle does not take up any space. Well, my usual overkill option of using tikz provides a method to place text without influencing the positioning.

You mark the two end points of where you want the text centered via a \tikzmark and give each of these marks a name. Then after the table you can place text anywhere relative to these marked points. Below I have set the default (show in red) placement below the title, but you can also use the first optional parameter to the \AddSubTitle macro to tweak the positioning and color as I have done in the second example (text in blue):

I am not sure that you would want to place text on top of the \tabucline, but that might make sense for an ornament. Anyway you have the flexibility to place it where you want.

Perhaps something like this first example below might be useful where I have added a white fill around the text so that the subtitle is not obscured:

Notes:

• This does require two runs. First one to determine the locations, and the second to do the drawing.

• The \tikzmark is from Adding a large brace next to a body of text.

Code:

\documentclass{article}
\usepackage{tabu} 
\usepackage{tikz}
\usetikzlibrary{calc}

\newcommand{\tikzmark}[1]{\tikz[overlay,remember picture] \node (#1) {};}
\newcommand*{\AddSubTitle}[4][]{%
    % #1 = draw options
    % #2 = left point
    % #3 = right point
    % #4 = text
    \begin{tikzpicture}[overlay,remember picture]
        \coordinate (MidPoint) at ($(#2.south)!0.5!(#3.south)$);
        \node [red, #1] at (MidPoint)  {#4};
    \end{tikzpicture}%
}%

\newcommand{\MyTabu}{%  Make it easier to repeat for testing purposes
\noindent
\begin{tabu}{|X[m,c]|X[-2,m,c]|X[m,c]|} \tabucline \\
    \scriptsize This is something \linebreak on the left
    &
    \Huge \tikzmark{left}Here's a Title\tikzmark{right}
    &
    \scriptsize And here's some stuff \linebreak on the \linebreak right side
    \\ \tabucline \\
\end{tabu}
}%
\begin{document}

\MyTabu
\AddSubTitle{left}{right}{And a subtitle!}

\vspace*{1.0cm}
\MyTabu
\AddSubTitle[blue, yshift=0.2cm]{left}{right}{And a subtitle!}

\vspace*{1.0cm}
\MyTabu
\AddSubTitle[red, yshift=0.5cm, rotate=45]{left}{right}{And a subtitle!}

\end{document}

Put the subtitle in a centred multicolumn spanning the three columns (I put a \strut after the huge title to have better spacing):

\documentclass{article}
\usepackage{tabu} 

\begin{document}

\noindent
\begin{tabu}{|X[m,c]|X[-2,m,c]|X[m,c]|} \tabucline \\
    \scriptsize This is something \linebreak on the left
    &
    \Huge Here's a Title\strut
    &
    \scriptsize And here's some stuff \linebreak on the \linebreak right side
    \\\tabucline\\
\multicolumn{3}{|c|}{Her is the subtitle}\\ \tabucline \\
\end{tabu}

\end{document}

Or just in a separate row (experiment with \strut after the subtitle to see if you find the spacing better):

\documentclass{article}
\usepackage{tabu} 

\begin{document}

\noindent
\begin{tabu}{|X[m,c]|X[-2,m,c]|X[m,c]|} \tabucline \\
    \scriptsize This is something \linebreak on the left
    &
    \Huge Here's a Title\strut
    &
    \scriptsize And here's some stuff \linebreak on the \linebreak right side
    \\
& Her is the subtitle \strut &\\ \tabucline \\
\end{tabu}

\end{document}

Your third option is to use the package textpos. By placing :

\documentclass{article}
\usepackage{textpos} 
\usepackage{tabu}

\setlength{\TPHorizModule}{\textwidth}
\setlength{\TPVertModule}{\textheight}


\begin{document}

\noindent
\begin{tabu}{|X[m,c]|X[-2,m,c]|X[m,c]|} \tabucline \\
    \scriptsize This is something \linebreak on the left
    &
    \Huge Here's a Title\strut
    &
    \scriptsize And here's some stuff \linebreak on the \linebreak right side
 \\ \tabucline \\
\end{tabu}

\begin{textblock}{0.8}[0.5,0.5](0.5,-0.01)
\noindent\centering Subtitle
\end{textblock}

\end{document}