The link I gave earlier was to a KwiKFit website, which automatically makes people sceptical. The following link is to a RoSPA site:
http://www.rospa.com/road-safety/advice/vehicles/tyre-safety-technology/information/#aging
I also found a quote after Googling "tyre aging" - Carmakers such as Nissan and Mercedes-Benz tell consumers to replace tires six years after their production date, regardless of tread life. Tire manufacturers such as Continental and Michelin say a tire can last up to 10 years, provided you get annual tire inspections after the fifth year.
It's certainly worth checking the manufacture date on your tyres since they will certainly be older than your car, maybe by over a year.
There's only your brakes, tyres and steering between you and the undertaker.