Five teams took part in putting out the blaze at the site, which covers 3,000 square metres, the Kuwait Fire Force said.
The department said later on Friday that the fire had been extinguished "without injuries".
In 2021, Kuwait's government removed all of the tyres from a dump near Mutlaa to Al Salmi, near the Saudi border, where recycling efforts had begun.
Scrap tyres are a major environmental problem worldwide due to their bulk and the chemicals they release.
The government has expressed hope that Al Salmi would become a tyre-recycling hub, with more factories planned for the future.
At a plant run by the Epsco Global General Trading recycling company, employees sort and shred scrap tyres before pressing the particles into rubbery coloured flooring tiles.
Set up in 2019, the recycling plant stands on a 50,000-square-metre lot near Al Salmi tyre dump.