The children were among 19 people - including 10 other children - caught in the inferno at the Gympanzee nursery in the Villaggio shopping mall on May 28, 2012.
"This is Coca-Cola, it's an organisation that understands all about the power of brand and Coca-Cola is based in Atlanta in Georgia, one of Lillie, Jackson and Willsher's friends who was killed in the fire came from Atlanta so it's even knocking down a kid in their own backyard."
The Weekes, who gave birth to twins Poppy and Parker in August, complained to Coca-Cola yesterday, but had not had a response last night.
However, the company did remove the signage early this morning. In a Facebook post, Mr Weekes thanked Coca Cola for its "prompt and proactive" response.
But Mr Weekes also said every new issue that arose related to the fire caused more distress for the Devonport family.
Following the fire, blamed on an electrical fault, five people including the owners of the nursery, members of mall management and a government official were convicted of involuntary manslaughter and sentenced to jail in July.
Appeal processes meant those convicted were yet to serve their sentences and the Weekes were embroiled in the ongoing legal action in Qatar.
"At this point in time we just seem to be spending our lives and all the money we have chasing a list of people who just don't seem to want to be held accountable.
"If they think we are going to forget about our children they are wrong, and other people aren't going to stop either. It didn't take long before the poster went out for people to start contacting me so obviously a lot of people are upset by it.
"Keeping the story out there and the pressure on is the only way to get justice served."
Coca-Cola said in a statement to the Herald: "We sincerely apologise for the placement of this ad. This panel in the neighbouring area of the mall was part of a network of outdoor locations where we ran the ad. It was not intended to be placed specifically near the site of the tragedy. We will ensure we have a more thorough review process in place going forward. The ad is being removed immediately."