\textbf{Step 1: Understand the Formation of $ \text{B_2\text{O}_3 $}}
$ \text{B}_2\text{O}_3 $ is a common oxide of boron and is typically formed when boron or its compounds react with oxygen under certain conditions. Let's analyze each option to determine whether $ \text{B}_2\text{O}_3 $ is formed.
Step 2: Analyze Each Option
Option (1): Acidifying an aqueous solution of $ \text{Na}_2\text{B}_4\text{O}_7 $
$ \text{Na}_2\text{B}_4\text{O}_7 $ is sodium tetraborate, also known as borax.
When acidified, the reaction produces boric acid ($ \text{H}_3\text{BO}_3 $), not $ \text{B}_2\text{O}_3 $. The reaction is:
$$
\text{Na}_2\text{B}_4\text{O}_7 + 2\text{HCl} \rightarrow 4\text{H}_3\text{BO}_3 + 2\text{NaCl}
$$
No $ \text{B}_2\text{O}_3 $ is formed.
Option (2): Heating of Boron in air
When boron is heated in air, it reacts with oxygen to form $ \text{B}_2\text{O}_3 $. The reaction is:
$$
4\text{B} + 3\text{O}_2 \rightarrow 2\text{B}_2\text{O}_3
$$
$ \text{B}_2\text{O}_3 $ is formed.
Option (3): Burning of $ \text{B}_2\text{H}_6 $ in oxygen
$ \text{B}_2\text{H}_6 $ (diborane) burns in oxygen to form $ \text{B}_2\text{O}_3 $ and water. The reaction is:
$$
\text{B}_2\text{H}_6 + 3\text{O}_2 \rightarrow \text{B}_2\text{O}_3 + 3\text{H}_2\text{O}
$$
$ \text{B}_2\text{O}_3 $ is formed.
Option (4): On heating $ \text{H}_3\text{BO}_3 $ above $ 370 \, \text{K} $
When boric acid ($ \text{H}_3\text{BO}_3 $) is heated above $ 370 \, \text{K} $, it decomposes to form $ \text{B}_2\text{O}_3 $ and water vapor. The reaction is:
$$
2\text{H}_3\text{BO}_3 \xrightarrow{\text{above } 370 \, \text{K}} \text{B}_2\text{O}_3 + 3\text{H}_2\text{O}
$$
$ \text{B}_2\text{O}_3 $ is formed.
Step 3: Identify the Correct Option
From the analysis, Option (1) is the only process where $ \text{B}_2\text{O}_3 $ is not formed.