From Thomas Jefferson Encyclopedia
It is unclear if Jefferson ever played billiards. He writes in Thoughts on Lotteries: "...But there are some [games of chance] which produce nothing, and endanger the well-nbeing of the individuals engaged in them, or of others depending on them. Such are games with cards, dice, billiards, etc."
Billiard Table at Monticello
The story goes that Jefferson did not install a billiards table in there because a Virginia law either taxed it or made it a misdemeanor for private residences to do so. You see some reference to billiards and the dome in the 1830s including one by William Waller Hening in The Statutes at Large: "The Second floor contains about fourteen rooms and numerous closets with a large circular Billiard Saloon under the Rotunda with Circular Windows."
However, there is no reference to the dome room being used as a billiards room by Jefferson. In Merrill Peterson's Visitors to Monticello, he states, "...nor was any law banning billiards passed in Virginia, nor did Jefferson intend the dome for billiards, nor was it converted into a ballroom."