The correct answer is $\boxed{\text{C}}$, 2,3-hexadiene.
(E)-3-bromo-3-hexene is a conjugated alkene. When treated with a strong base, such as $\ce{CH3O-}$ in $\ce{CH3OH}$, it undergoes an E2 elimination reaction to form a conjugated diene. The elimination reaction proceeds via a concerted mechanism, with the proton and the leaving group departing in the same step. The product of the elimination reaction is 2,3-hexadiene, which is a conjugated diene. Conjugated dienes are more stable than isolated double bonds, and therefore the elimination reaction is favored.
The other options are incorrect because they do not represent the product of an E2 elimination reaction. Option A, 3-hexyne, is an alkyne. Alkynes are less stable than alkenes, and therefore the elimination reaction would not produce an alkyne. Option B, 2-hexyne, is also an alkyne. Option C, 2,3-hexadiene, is a conjugated diene, but it is not the product of an E2 elimination reaction. Option D, 2,4-hexadiene, is a conjugated diene, but it is not the product of an E2 elimination reaction.