Chọn phương án C.

\(\begin{eqnarray*}
&\log_3x\cdot\log_9x\cdot\log_{27}x\cdot\log_{81}x&=\dfrac{2}{3}\\
\Leftrightarrow&\log_3x\cdot\log_{3^2}x\cdot\log_{3^3}x\cdot\log_{3^4}x&=\dfrac{2}{3}\\
\Leftrightarrow&\log_3x\cdot\dfrac{1}{2}\log_3x\cdot\dfrac{1}{3}\log_3x\cdot\dfrac{1}{4}\log_3x&=\dfrac{2}{3}\\
\Leftrightarrow&\dfrac{1}{24}\left(\log_3x\right)^4&=\dfrac{2}{3}\\
\Leftrightarrow&\left(\log_3x\right)^4&=16\\
\Leftrightarrow&\left[\begin{array}{l}\log_3x=2\\ \log_3x=-2\end{array}\right.\Leftrightarrow\left[\begin{array}{l}x=9\\ x=\dfrac{1}{9}\end{array}\right.
\end{eqnarray*}\)

Khi đó tổng các nghiệm cần tìm là $$9+\dfrac{1}{9}=\dfrac{82}{9}$$