Suppose that $a$ and $b$ are positive real numbers such that $3\log_{101}\left(\frac{1,030,301-a-b}{3ab}\right) = 3 - 2 \log_{101}(ab)$. Find $101 - \sqrt[3]{a}-  \sqrt[3]{b}$.
1. Start with the given equation:
   \[
   3\log_{101}\left(\frac{1,030,301 - a - b}{3ab}\right) = 3 - 2 \log_{101}(ab)
   \]

2. Divide both sides by 3:
   \[
   \log_{101}\left(\frac{1,030,301 - a - b}{3ab}\right) = 1 - \frac{2}{3} \log_{101}(ab)
   \]

3. Let \( x = \log_{101}(ab) \). Then the equation becomes:
   \[
   \log_{101}\left(\frac{1,030,301 - a - b}{3ab}\right) = 1 - \frac{2}{3} x
   \]

4. Rewrite the right-hand side using properties of logarithms:
   \[
   \log_{101}\left(\frac{1,030,301 - a - b}{3ab}\right) = \log_{101}(101) - \log_{101}((ab)^{2/3})
   \]

5. Since \(\log_{101}(101) = 1\), we have:
   \[
   \log_{101}\left(\frac{1,030,301 - a - b}{3ab}\right) = \log_{101}\left(\frac{101}{(ab)^{2/3}}\right)
   \]

6. Equate the arguments of the logarithms:
   \[
   \frac{1,030,301 - a - b}{3ab} = \frac{101}{(ab)^{2/3}}
   \]

7. Cross-multiply to clear the fraction:
   \[
   (1,030,301 - a - b)(ab)^{2/3} = 303ab
   \]

8. Let \( \sqrt[3]{a} = x \) and \( \sqrt[3]{b} = y \). Then \( ab = x^3 y^3 \) and the equation becomes:
   \[
   (101^3 - x^3 - y^3)(x^2 y^2) = 303x^3 y^3
   \]

9. Simplify the equation:
   \[
   101^3 x^2 y^2 - x^5 y^2 - x^2 y^5 = 303x^3 y^3
   \]

10. Factor out \( x^2 y^2 \):
    \[
    101^3 - x^3 - y^3 = 303xy
    \]

11. Let \( s = x + y \) and \( p = xy \). Then:
    \[
    101^3 - s^3 + 3sp = 303p
    \]

12. Rearrange the equation:
    \[
    101^3 = s^3 - 3sp + 303p
    \]

13. Since \( s = 101 \), substitute \( s \) into the equation:
    \[
    101^3 = 101^3 - 3 \cdot 101 \cdot p + 303p
    \]

14. Simplify:
    \[
    0 = -303p + 303p
    \]

15. This confirms that \( s = 101 \) is correct. Therefore:
    \[
    101 - \sqrt[3]{a} - \sqrt[3]{b} = 101 - s = 101 - 101 = 0
    \]

The final answer is \(\boxed{0}\).