What will be the major product on chlorination of m-nitrophenol?
But why is the activating nature of hydroxyl group dominating over the deactivating nature of Nitro group?   


This is the precise reason why Picric acid is formed. ( Check here )
The activating nature of the $\ce{-OH}$ group is strong enough to counter the deactivating nature of the nitro groups.
Look at Ron's comment on the question over here.
Even methyl group sustains deactivating nature of nitro group in the formation of TNT. So no wonder phenolic group, a better activator, dominates.  

The phenolic group increases electron density on ortho para (making it o-p director) while nitro groups decrease e-density on the o-p positions (making it a meta director). When both these effects act simultaneously, the question is who wins and is the density on o-p carbons less than corresponding meta ones? (as that would decide where the electrophile will attack)  

Resonace structures of the groups gives the solution.
In nitro group:  

    

There are three charged atoms bonded together in the $\ce{-NO_2}$ molecule. This is less table than the positive charge bearing oxygen atom in phenolic group's resonance structure: 

    

Hence phenolic group wins!!  

Coming back to the given starting reactant:     

   

So now we consider the dominating ortho-para effect of phenolic group over the meta-directing effect of nitro groups to get 4-chloro-3-nitrophenol: