The only EU country which has chosen to remain outside of the full provisions of the Schengen Agreement is the United Kingdom. Ireland is forced to remain outside of Schengen because of its 'open border' agreement with the UK.
For anyone to enter the Schengen block, they need to show a valid passport. The only exception to that rule is that a national identity card may be used by EU citizens. (A driving licence is not a national identity card).
So, until the UK introduces ID cards, UK citizens require a passport to enter any other country, with the exception of Ireland. (UK citizens do not require a passport to enter Ireland but they do require ID to meet the carrier's security rules. All airlines and ferry companies accept a photo-style licence as suitable ID).
Chris