Let $r$, $s$, and $t$ be the three roots of the equation
\[8x^3 + 1001x + 2008 = 0.\]
Find $(r + s)^3 + (s + t)^3 + (t + r)^3$.
By [Vieta's formulas](https://artofproblemsolving.com/wiki/index.php/Vieta%27s_formulas), we have $r + s + t = 0$ so $t = -r - s.$ Substituting this into our problem statement, our desired quantity is \[(r + s)^3 - r^3 - s^3 = 3r^2s + 3rs^2 = 3rs(r + s).\] Also by [Vieta's formulas](https://artofproblemsolving.com/wiki/index.php/Vieta%27s_formulas) we have \[rst = -rs(r + s) = -\dfrac{2008}{8} = -251\] so negating both sides and multiplying through by 3 gives our answer of $\boxed{753}.$