Can terminal enols be reduced by using LiAlH4? According to the solution given in my textbook, no reaction takes place in this case. 

But I am not quite sure about it. I feel that the enol can easily tautomerize to aldehyde and we know that aldehydes are reduced by $\ce{LiAlH4}$ to give alcohols.

So, will the above reaction take place? What will be the final product after reduction occurs?
Not only can that enol easily tautomerize to the aldehyde, the aldehyde is the major component at equilibrium at room temperature (by a lot!). The equilibrium constant for the similar system of ethanal/vinyl alcohol (one carbon shorter than yours) is $K \approx 3\times 10^{-7}$ (source: Wikipedia page on keto-enol tautomerism):

$$\ce{ CH3CHO &lt;=&gt; CH2=CHOH}\\
K = \dfrac{[\ce{CH2=CHOH}]}{[\ce{CH3CHO}]}\approx 3\times 10^{-7}$$ 

Thus, for every molecule of enol in this system, there are $\sim 3 \times 10^6$ molecules of aldehyde. $\ce{LiAlH4}$ will reduce the aldehyde and then LeChâtelier's principle will drag the enol along for the ride.