Why 3-chloroprop-1-en-1-ol is unstable?
3-Chloroprop-1-en-1-ol is in equilibrium with the aldehyde, 3-chloropropanal (aka $\beta$-chloropropionaldehyde).

According to Organic Syntheses, this compound "is a very unstable substance which polymerizes rapidly, especially in the presence of traces of hydrochloric acid".

But I do not see it being reduced to 3-chloropropanol.



Reactivity/instability

In 3-chloropropanal, or more precisely in its enol form 3-chloropropenol, the chlorine atom is in allylic position (i.e. on a carbon atom adjacent to a $\ce{C=C}$ double bond) which makes it more reactive towards nucleophilic substitution. Another molecule of the enol can then substitute it and the resulting product is

$\ce{Cl-CH2-CH=CH-O-CH2-CH=CH-OH}$

which still has an allylic chlorine atom, so it can react further, hence the polymerization.