Adding shadow to font in beamer title

A quick hack would be to write the title a second time, a bit offset and in a darker colour. The following example uses a mixture of the background colour and black \color{black!20!green-abs} and is a bit darker than the shadow shown in your example, but this can be adjusted to fir your needs.

\setbeamertemplate{frametitle}{
    \begin{beamercolorbox}[wd=\paperwidth, ht=1.5cm]{frametitle}
        \begin{tikzpicture}[remember picture, overlay]
        \shade[outer color=green-abs, inner color=light-green-abs]
        (0, 0) rectangle (\the\paperwidth, 1.55cm);
        %%%%%%
        % I'd like to set the style of frametitles
        %%%%%%
        \node[anchor = west, titlecolor-abs] at (0.53, 0.77)
        {\usebeamerfont{frametitle}\color{black!20!green-abs}\insertframetitle};
        \node[anchor = west, titlecolor-abs] at (0.5, 0.8)
        {\usebeamerfont{frametitle}\insertframetitle};
        \end{tikzpicture}
    \end{beamercolorbox}
}

Edit:

And for a smoother shadow, one could stack more layers of different shades. In the following example 5 layers are used to create a smoother shadow:

\setbeamertemplate{frametitle}{
    \begin{beamercolorbox}[wd=\paperwidth, ht=1.5cm]{frametitle}
        \begin{tikzpicture}[remember picture, overlay]
        \shade[outer color=green-abs, inner color=light-green-abs]
        (0, 0) rectangle (\the\paperwidth, 1.55cm);
        %%%%%%
        % I'd like to set the style of frametitles
        %%%%%%
        \foreach \x in {0,5,...,20}
            \node[anchor = west, titlecolor-abs, xshift=\x*-0.002cm, yshift=\x*0.002cm] at (0.5, 0.8) {%
                \usebeamerfont{frametitle}%
                \color{black!\x!green-abs}%
                \insertframetitle%
            };
        \node[anchor = west, titlecolor-abs] at (0.5-0.002*20, 0.8+0.002*20)
        {\usebeamerfont{frametitle}\insertframetitle};
        \end{tikzpicture}
    \end{beamercolorbox}
}

I'll complete samcarter's answer with a couple modifications I added: I moved the code to a function, and I darkened the shadow.

Function:

\tikzset{
    text shadow/.code args={[#1]#2at#3(#4,#5)#6}{
        \pgfkeysalso{/tikz/.cd,#1}
        \foreach \x in {0,10,...,40}{%
            \node[#1, xshift=\x*-0.001cm, yshift=\x*0.001cm] at (#4,#5){%
            \color{black!\x!green-abs}
            #6
            };
        }
        \node[#1] at (#4-0.001*40, #5+0.001*40){#6};
    }
}

(tzset example from here)

Frametitle:

\setbeamertemplate{frametitle}{
    \begin{beamercolorbox}[wd=\paperwidth, ht=1.5cm]{frametitle}
    \begin{tikzpicture}[remember picture, overlay]
        \shade[outer color=green-abs, inner color=light-green-abs]
            (0, 0) rectangle (\the\paperwidth, 1.55cm);
        \node[text shadow={[anchor = west, titlecolor-abs] at (0.5, 0.8)
            {\usebeamerfont{frametitle}\insertframetitle}}] {};
    \end{tikzpicture}
    \end{beamercolorbox}
}

Title page:

\setbeamertemplate{title page}{
    \vfill
    \centering
    \begin{beamercolorbox}[sep=25pt, center]{title}
        \begin{tikzpicture}[remember picture, overlay]
        \node[text shadow={[titlecolor-abs, text width=0.85\paperwidth, text centered] at (0, 0)
            {\usebeamerfont{title}\inserttitle\par}}] {};
        \end{tikzpicture}
    \end{beamercolorbox}
    \vskip1em\par
    \begin{beamercolorbox}[sep=8pt,center]{author}
        \usebeamerfont{author}\insertauthor
    \end{beamercolorbox}
    \begin{beamercolorbox}[sep=8pt,center]{institute}
        \usebeamerfont{institute}\insertinstitute
    \end{beamercolorbox}
    \begin{beamercolorbox}[sep=8pt,center]{date}
        \usebeamerfont{date}\insertdate
    \end{beamercolorbox}\vskip0.5em
    {\usebeamercolor[fg]{titlegraphic}\inserttitlegraphic\par}
    \vfill
}

These would be the results for the title page:

It looks really good!!