Step 1: Use the iodoform test result.
A positive iodoform test indicates the presence of either:
\[
\text{(i) } \mathrm{CH_3CO{-}} \text{ group (methyl ketone), or}
\text{(ii) } \mathrm{CH_3{-}CHOH{-}} \text{ group (secondary alcohol)}.
\]
Step 2: Analyze the oxidation behavior.
On oxidation, compound \(A\) forms compound \(B\) which does not reduce Tollens' reagent.
This means \(B\) is not an aldehyde, but a ketone, since aldehydes reduce Tollens' reagent while ketones do not.
Step 3: Deduce the nature of compound \(A\).
Since oxidation of \(A\) gives a ketone, compound \(A\) must be a secondary alcohol.
Step 4: Match with the molecular formula \( \mathrm{C_4H_8O} \).
Among the given options, butan-2-ol is a secondary alcohol with the group \( \mathrm{CH_3{-}CHOH{-}} \), satisfies the iodoform test, and oxidizes to a ketone (butan-2-one).
Step 5: Final conclusion.
Therefore, the correct compound \(A\) is:
\[
\boxed{\text{Butan-2-ol}}.
\]