Which of the following compounds is resistant to oxidation by periodic acid (HIO₄)?

Periodic acid cleaves compounds that contain a vicinal diol, or an $\alpha$-hydroxy carbonyl system that can form the required cleavage intermediate. Option $\mathrm{HOCH_2CH_2OH}$ has two neighboring $\mathrm{-OH}$ groups on adjacent carbons, so it is oxidized. Option $\mathrm{HOCH_2CH(OH)CH_2OH}$ contains adjacent alcohol groups, so it is oxidized. Option $\mathrm{HOCH_2COCH_2OH}$ is an $\alpha$-hydroxy ketone equivalent on both sides of the carbonyl and undergoes periodic acid oxidation. Option $\mathrm{HOCH_2CH_2CH_2OH}$ is a $1,3$-diol, so the hydroxyl groups are not on adjacent carbons and it resists oxidation by periodic acid. Therefore the correct option is $\mathrm{HOCH_2CH_2CH_2OH}$, which matches option $\mathrm{B}$.