This poem is about a couple of things.
First of all, it is about how things that are famous and important at one point come to be dead and irrelevant later on. It is the idea that even the famous and important die and become nothing.
Second, this is a poem about Irish nationalism. This poem was written at a time when England ruled Ireland. The poem uses the ancient seat of Irish kings -- Tara -- as a symbol for Irish independence. It is saying that Irish independence will survive as long as "some heart indignant breaks."
"The Harp that once through Tara's Halls" like many of Moore's Irish Melodies is a Lament. Tara which was the dwelling place of the kings of Ireland. And to this day the spiritual heart of Ireland. Moore compares the Harp of antiquity; a glorious, shimmering past contrasting with a wretched and dispossed present. The harp is now mute as Ireland is mute. The only glimmer of hope being, that when someone pause's and reflects on what has been lost can we dream of a future more in keeping with the past.