He worked for the preservation of the Iraqi Jewish rite and the opinions of the Ben Ish Chai, and opposed the attempts of R. Ovadia Yosef to impose a uniform "Israeli Sephardi" rite based on the Shulhan Arukh and his own halakhic opinions. He published a prayer book called Kol Eliyahu based on this stance.

R. Eliyahu is one of the spiritual leaders of the Religious Zionist movement and was an outspoken opponent of the Gaza Disengagement of 2005. He is considered somewhat controversial for his decades-long support of what some characterize as the radical right of the Religious Zionist movement. R. Eliyahu was a friend of R. Meir Kahane and his family. He officiated at the marriage of Binyamin Ze'ev Kahane and delivered the eulogy at R. Meir Kahane's funeral. He is a longtime supporter of Jonathan Pollard and became his spiritual mentor while Pollard was in prison.