Step 1: Analyze Reaction I: $\text{Reaction of sodium borohydride with iodine$.}
$\text{NaBH}_4$ is a source of hydride ions ($\text{H}^-$). The reaction is a synthesis of diborane, which typically evolves hydrogen gas, especially when reacting with an oxidizing agent like iodine ($\text{I}_2$).
\[
2\text{NaBH}_4 + \text{I}_2 \xrightarrow{\text{ether}} \text{B}_2\text{H}_6 + 2\text{NaI} + \text{H}_2 \uparrow
\]
Hydrogen gas is evolved. So, I is correct.
Step 2: Analyze Reaction II: $\text{Oxidation of diborane$.}
Diborane ($\text{B}_2\text{H}_6$) is oxidized (burned) in the presence of oxygen ($\text{O}_2$). The reaction is highly exothermic.
\[
\text{B}_2\text{H}_6 + 3\text{O}_2 \to \text{B}_2\text{O}_3 + 3\text{H}_2\text{O}.
\]
No hydrogen gas is evolved; water vapor is produced. So, II is incorrect.
Step 3: Analyze Reaction III: $\text{Reaction of boron trifluoride with sodium hydride$.}
This reaction is a laboratory method for the preparation of diborane ($\text{B}_2\text{H}_6$). It is a reduction reaction.
\[
2\text{BF}_3 + 6\text{NaH} \xrightarrow{450 \text{ K}} \text{B}_2\text{H}_6 + 6\text{NaF}.
\]
No hydrogen gas is evolved. So, III is incorrect.
Step 4: Analyze Reaction IV: $\text{Hydrolysis of diborane$.}
Diborane reacts with water (hydrolysis) to form boric acid and hydrogen gas.
\[
\text{B}_2\text{H}_6 + 6\text{H}_2\text{O} \to 2\text{H}_3\text{BO}_3 + 6\text{H}_2 \uparrow
\]
Hydrogen gas is evolved. So, IV is correct.
Step 5: Conclude the final correct option.
Hydrogen gas is evolved in reactions I and IV.