How can I determine the date of manufacture of drywall?
Usually the year the house was built is a good way to start, but if the drywall was added as a renovation, the date stamp is on the back side of the sheets. If you can get access, say in an attic space and move insulation if needed will get you there. If the back is not accessible at all, then removal of a big enough piece will show you that. Mind you, the date stamp is not all over the place, it will only be along one edge or perhaps even in the middle. 
There are other ways to get an idea how old it may be if you can determine if other work was done at the same time, for example, doors made in a big facility will have a date stamp on the top or bottom of the leaf, IF it hasn't been cut off or painted over. 

I am going to hazard a guess you want to know because of asbestos?
FWIW, I hear that sheetrock produced in Canada still has asbestos in it since it was just banned in recent years. Just like in the States when it was banned in the late 70's they allowed it to be sold until the early 80's so the stock piles would get sold. How's that for the power of marketing and capitalism. 

About the Canadian source, none of this is substantiated, it is something I heard of in passing since I am now in the Northwest and happen to be in the construction trade.