About 40,000 users downloaded Firechat last week, compared with 6,600 over the previous few months, the company says.
The internet has been blocked in some Iraqi provinces, as authorities seek to prevent militants from communicating.
Access to social media sites has also been severely restricted.
Firechat allows users to take part in group chats with between two and 10,000 people, without the need for an internet connection.
Using a technology known as "mesh networking", messages can be sent to people within the immediate vicinity, as long as they too have the app installed. However, discussions are not private, and can be seen by anyone in the area.
The software is available for both Android and iOS devices, and has a range of roughly 70m (230ft). However, if enough people use the app, messages can travel over far greater distances, hopping between intermediary devices in a chain-like effect.
The app was heavily used in Taiwan earlier this year, when protesting students intent on occupying the parliament  were faced with the threat of internet restrictions and limited cell coverage.
Firechat does not have access to the content of the messages.
Over the past couple of weeks, Iraqis attempting to visit social media sites have been greeted by a message saying the Ministry of Communications has barred access.
The government has also ordered the internet to be completely shut down in some provinces, where Isis militants are active.
The move was taken after Islamist insurgents used Twitter to post a graphic image of a beheaded man, and to spread propaganda messages.
Firechat, which was launched three months ago by Californian firm Open Garden, says the app's popularity in Iraq is now second only to the US.
A spokesperson for the company said the number of users in Iraq might have been underestimated, as many were using virtual private networks (VPNs), which disguise activity, to access the app.
Psiphon, a system which allows users to circumvent internet censorship, told the BBC it had seen a "huge influx" in the numbers of those using its service in Iraq.
Iraqis have been turning to an app which allows group messages to be sent between phones, without the need for an internet connection, in an effort to circumnavigate government restrictions.