Step 1: Understand the question.
We must find the pair that are position isomers. Position isomers have the same molecular formula and the same carbon skeleton, but the double bond sits in a different place.
Step 2: Check but-1-ene and 2-methylprop-1-ene.
Both are $\text{C}_4\text{H}_8$, but one is a straight chain and the other is branched. Different skeletons make them chain isomers, not position isomers.
Step 3: Check but-1-ene and 2-methylbut-1-ene.
These have different formulas, $\text{C}_4\text{H}_8$ and $\text{C}_5\text{H}_{10}$. Different formulas means they are not isomers at all.
Step 4: Check but-2-ene and 2-methylprop-1-ene.
These have the same formula $\text{C}_4\text{H}_8$ but different skeletons (straight versus branched), so they are chain isomers, not position isomers.
Step 5: Check but-1-ene and but-2-ene.
Both are $\text{C}_4\text{H}_8$ with the same straight four carbon chain. The only difference is the double bond at carbon 1 versus carbon 2. That is exactly position isomerism.
Step 6: Pick the answer.
The position isomers are but-1-ene and but-2-ene, which is option 3.
\[ \boxed{\text{But-1-ene and but-2-ene}} \]