It's a slight exaggeration, but the core appears to be true.
Wikipedia has a page on the Hitler family and what happened to it, which sources a lot of its information from The Last of the Hitlers by David Gardner. In that book, he tracks down and interviews the surviving members of the family.
Hitler had a sister, a half-brother, and a half-sister who survived into adulthood and could have had children. None of them lived past 1960, although both half-siblings had children. He also had two sister-in-laws. One sister-in-law never married and never had any kids. The other had one daughter, who committed suicide in 1971 (for unrelated reasons).
It's highly unlikely there was any sort of "pact" in this generation - all Hitler's siblings (of all sorts) had children or not based on their own lives, and wouldn't have that decision affected by his reputation.
As previously mentioned, Hitler had a niece through Eva Braun who committed suicide before having any kids. The more interesting (and relevant) people are the five children between his two half-siblings.
Wikipedia (again) summarizes it as follows:
Of [his half-sister's] children, Leo Rudolf Raubal Jr had one son, Peter Raubal, in 1931; Elfriede Raubal married Dr. Ernst Hochegger in 1937 and had a son, Heiner Hochegger, in 1945; and Geli Raubal committed suicide without issue in 1931. Both Peter and Heiner have had no children.
[Hitler's half-brother,] William Patrick, married Phyllis Jean-Jacques in 1947 in the US, where they had four children. Alexander Adolf Stuart-Houston (1949), Louis Stuart-Houston (1951), Howard Ronald Stuart-Houston (1957), and Brian William Stuart-Houston (1965) have all had no children; none of them married, with the exception of Howard, who died in a car crash in 1989.
So was there a pact?
So the question now is whether Peter Raubal, Heiner Hochegger, and Alexander, Louis, Howard and Brian Stuart-Houston had a pact not to have children. This is where the book by David Gardner, and an accompanying article come in.
The Stuart-Houstons (who live in New York) likely have had no connection to their cousins Peter and Heiner (who live in Europe), as their father very much wanted to distance himself from his family. Peter is also in his 80's and Heiner is in his late 60's. As neither of them have had children, it's possible they independently decided not to have any, but it's unlikely it was part of any sort of pact.
Finally, that leaves the three living Stuart-Houstons - the ones most likely to have made a pact. Gardner writes:
None of the three sons has married, and there are no children. Alex initially denied that there had been a pact between the brothers to ensure that the Hitler line was not continued. Then he told me: "Maybe my other two brothers did [make a pact], but I never did." It was just one more contradiction to add to the many that already cloud his family history.
According to another source, he also wrote that
They didn’t sign a pact, but what they did is, they talked amongst themselves, talked about the burden they’ve had in the background of their lives, and decided that none of them would marry, none of them would have children. And that’s… a pact they’ve kept to this day.
So it's likely that while there is no formal "pact not to have children", all the siblings agreed that they didn't want to burden a child with their background. In other words, it's not a pact to make the family die out, it's a mutual understanding that any children they may have would have a very hard life.