Stand Up America Now made the threat after it emerged last week that the Iranian courts may have upheld the execution order for Pastor Youcef Nadarkhani.
The pastor has been in prison since October 2009. He was originally detained for protesting but the charge was later changed to apostasy.
Dr Jones sparked an international panic in 2010 when he threatened to burn Korans on the anniversary of 9/11.
After sparking riots in parts of the Middle East and Asia, the pastor came under intense pressure from political and religious leaders, and agreed not to go ahead with the stunt.
No sooner was he out of the headlines when his small Florida church burned a Koran in March last year after putting the holy book on trial and finding it guilty of crimes against humanity.
That stunt triggered massive riots in the northern Afghanistan city of Mazar-i-Sharif, where 10 UN officials were killed in an attack on the organisation's city base.
Dr Jones defended his organisation's latest plans to The Christian Post, saying that Christians "cannot just stand by and do nothing" in the face of militant Islam.
"We have very little that we can do as a Christian community to protest the inhumane activities of Islam," Jones told the website.
"It's a type of protest to let the government know we are totally disgusted with their type of Shariah, with their activities. In fact, with the activities of Islam over the past 1,400 years."
Dr Jones has voiced approval for the burning of Korans in Afghanistan by members of the US military, which prompted riots that killed two Americans and several Afghans.
Jordan Sekulow, executive director of the American Center for Law and Justice, criticised Dr Jones' plans.
"It's an abuse of freedom," he told The Christian Post.
"We have those kinds of freedoms here, but if you're going to do it for publicity's sake, it still doesn't make it right just because it's legal."