Lucas reagent is a solution of zinc chloride in concentrated hydrochloric acid (ZnCl$_2$ in HCl) and is used to classify alcohols based on their reactivity. Alcohols react with Lucas reagent to form alkyl chlorides, with the reaction proceeding via a carbocation intermediate.

The reactivity order with Lucas reagent is:

Tertiary alcohols: React immediately at room temperature.

Secondary alcohols: React slowly.

Primary alcohols: Do not react at room temperature.

Analyzing the options:

Option A: H$_2$C = CHCH$_2$OH

This compound is allylic alcohol, which is generally more reactive than primary alcohols due to the stabilization of the carbocation by resonance. However, it still behaves like a primary alcohol with respect to Lucas reagent, which means it will not react at room temperature.

Option B: C$_6$H$_5$CH$_2$OH

This is benzyl alcohol. While benzyl alcohol is a primary alcohol, the benzyl group can stabilize the carbocation formed during the reaction, making it more reactive than other primary alcohols. Benzyl alcohol can react with Lucas reagent.

Option C: CH$_3$CH$_2$CH$_2$OH

This compound is a primary alcohol (propan-1-ol). Primary alcohols do not react with Lucas reagent at room temperature.

Option D: (CH$_3$)$_3$COH

This is tert-butyl alcohol, a tertiary alcohol. Tertiary alcohols react immediately with Lucas reagent, even at room temperature.

Therefore, the compound that would not react with Lucas reagent at room temperature is:

Option C: CH$_3$CH$_2$CH$_2$OH (Propan-1-ol)