Dúnhere believes the Enemy many be using the beacons to scout for Rohirrim and asks of you to patrol the beacons for the safety of the riders.
'The men I command need to rest if we are to be battle-ready when Minas Tirith is within our sights. I am in need of someone to scout the nearby beacons of Gondor. It is imperative that the Enemy not know our location or plans.'
The beacons of Amon Dîn and Eilenach can be found in Taur Drúadan.
You should defeat any enemies gathered at these beacons.
Dúnhere can be found in the War-stead of the Helmingas.
You should speak with Dúnhere.