With British nobiliary titles, the pre-emenant title is the one that is used, not all of their titles. Listing him as "3rd Earl of Cork and 2nd Earl of Burlington" is contrary to convention. If that were not so, then say, Prince Charles, Duke of Cornwall would have more titles listed than could possibly fit in the names field, since he also happens to hold the titles of Prince of Wales, Duke of Chester, Earl of Rothsay, Earl of Carrick, Baron of Renfrew, Lord of the Isles, and Prince and Great Steward of Scotland.
Charles Boyle was known as "Lord Burlington", as the Earldom of Burlington was the superior title, because it was in the peerage of England, while the Earldom of Cork was in the peerage of Ireland, and thus inferior to England (which ruled Ireland at the time). It's also worth noting that Charles Boyle held many other titles, including Viscount Dungarvan (in the peerage of Ireland), and the Barony of Clifford of Lanesborough, in England.