Step 1: Recall the reaction of $NaNH_2$.
Sodamide $NaNH_2$ is a very strong base. It pulls off the acidic hydrogen attached to a triple-bonded carbon.
Step 2: Which alkynes have that acidic H.
Only terminal alkynes, of the form $R-C\equiv C-H$, have a hydrogen directly on the triple-bonded carbon. Internal alkynes $R-C\equiv C-R$ have no such hydrogen.
Step 3: Check each choice.
Ethyne $HC\equiv CH$ is terminal, propyne $CH_3-C\equiv CH$ is terminal, and but-1-yne $CH_3CH_2-C\equiv CH$ is terminal. All three react.
Step 4: Find the odd one out.
But-2-yne is $CH_3-C\equiv C-CH_3$, an internal alkyne with no hydrogen on the triple-bond carbons.
Step 5: Conclusion.
So but-2-yne has no reaction with $NaNH_2$. \[ \boxed{\text{But-2-yne}} \]