Holy Trinity Church on Goodramgate posted photos of the felled tree on Twitter ahead of its removal. Several gravestones can be seen beneath it.
Staff expect to re-open as normal on Tuesday.
The preserved Grade I-listed church dates from the 12th century, and worship was suspended between 1882 and 1937 due to the building's poor state of repair.
In 1834, Halifax landowner Anne Lister and her partner Ann Walker travelled from their home at Shibden Hall to take Holy Communion at Holy Trinity during Easter.
Although same-sex weddings were forbidden at the time, the couple considered the ceremony to have been a legally binding marriage and lived as spouses from then on.
Anne was familiar with York, having attended a girls' boarding school housed in King's Manor, now a university property. It was there that she enjoyed early lesbian relationships.
Anne's life was the subject of the recent BBC period drama Gentleman Jack, which was filmed at locations across Yorkshire and was a huge hit with audiences. The cast and crew used Holy Trinity to recreate the wedding ceremony and the church appears in the series. A blue plaque outside the chapel commemorates what is believed to have been Britain's first lesbian wedding.
Holy Trinity is now in the care of the Churches Conservation Trust, and has not been used for worship since the 1970s.