It's really a two sided question this. Loyalists have been painting murals for over a hundred years now, with the first being a portrait of William of Oranges victory at the Battle of the Boyne over Catholic forces (1690) in Belfast, with similar going up in Derry around 1920.

The oldest still in existence today is this one from 1934, although it has been touched up and repainted many times since then.

On the Catholic side, Murals are a lot newer. Had a catholic mural gone up in 1908, the British government of the time would have reacted.

It wasn't until the 1970s where the system for political prisoners from Northern Ireland affected the beginning of Catholic murals - the new prison at Long Kesh, the Maze, was set up specifically for paramilitary prisoners and encouraged non violent ways of protest amongst it's prisoners, particularly painting. The first mural went up in Belfast in 1981 as a resonse to the hunger strike at the prison, and with the release of many of the prisoners armed with brushs instead of bombs and the peace process many more went up.

This site appears to be a pretty good academic source on the history behind many of the more famous murals.