By catalysing hydrogen peroxide decomposition, it does react with it, but this reaction cyclically regenerate it to the initial state, with manganese reversibly switching oxidation state.
One of possible models is:
$$\begin{align}
\ce{H2O2(aq) + 2 MnO2(s) &-> H2O(l) + Mn2O3(s) + O2(g)}\\
\ce{H2O2(aq) + Mn2O3(s) &-> H2O(l) + 2 MnO2(s)}
\end{align}$$
The point is, hydrogen peroxide has both oxidative and reductive properties, with the latter being oxidized to oxygen.
Variants ( with the $\ce{MnO2}$ reduction part ongoing in the Leclanché cell) :
$$\begin{align}
\ce{H2O2(aq) + 2 MnO2(s) + H2O(l) &-> H2O(l) + 2 MnO(OH)(s) + O2(g)}\\
\ce{H2O2(aq) + MnO2(s) &-> Mn(OH)2(s) + O2(g)}\\
\ce{H2O2(aq) + 2 MnO(OH)(s) &-> 2 H2O(l) + 2 MnO2(s)}\\
\ce{H2O2(aq) + Mn(OH)2(s) &-> 3 H2O(l) + MnO2(s)}
\end{align}$$
Other model may involve also $\ce{Mn^2+(aq)}$ at low enough $\mathrm{pH}$.
The way of suppress the catalysis may be catching eventual intermediates e.g. by a reagent forming stable complexes with $\ce{Mn^{II}}$ or $\ce{Mn^{III}}$. It would cause majority of $\ce{Mn}$ ending in these complexes after oxidation enough $\ce{H2O2}$ to $\ce{O2}$.
Altenative would be using acidic environment where $\ce{MnO2}$ would end as $\ce{Mn^2+(aq)}$.
Reaction of $\ce{MnO2}$ with citric acid combines both, with acid undergoing oxidation and complexation.