Depends on the airline - some will accept a photo driving licence as proof of id.
bmibaby don't require a passport for domestic flights:
A passport is not required for travel between Great Britain and Ireland, Northern Ireland and the Channel Islands. However, passengers travelling with bmi are required to bring either a passport or some form of acceptable photographic identification when travelling between the UK and the Republic of Ireland. Please see acceptable forms of photographic ID, below
http://www.flybmi.com/bmi/en-gb/planning-and-i nformation/customer-service/booking-terms-and- conditions.aspx