It sort of depends on your definition of dating. But the closest that would come would be Woodrow Wilson. He was a recent widower and President when he met Edith Galt. They had a brief courtship where he pursued her pretty vigorously and they married in 1915.

There is also Grover Cleaveland. He was a bachelor when he entered office the first time and his sister acted as his first lady. However, he met and began a correspondence with a young lady, Frances Folsom, while in office. They married in the White House.

John Tyler's wife died while he was in office. And he later remarried, to Julia Gardiner, while he was still in office (remarried 1844 left office in 1845). It's hard to say what kind of courtship he had with his second wife prior though.

James Buchanan was never married (before, during or after his time in the White House). His niece acted as his first lady. There is certainly speculation concerning his personal life. There has been rumors that he was involved with William Rufus King, whom other democrats refereed to as his "wife" and "better half". But there is absolutely no way of confirming that.

Several other presidents have been single while in the White House but they were all widowers. Thomas Jefferson was the first single president (his wife, Martha, died in 1782 and he never remarried.) There is, of course, all sorts of speculation about his personal life but I don't think any of it would be considered "dating."