Do diapers have an expiration date?

Yes. But, it doesn't mean the product expires. Pampers says, "it doesn't mean the product expires...Paper products do not have a specific shelf life....We recommend our diapers be kept in a shaded, dry storage area where the temperature is 85 degrees or less. Extreme heat or humidity could effect their performance." they may change color, the glue get less sticky, gel may clump less as they get old.
None. As long as they are clean and have not gotten wet in the past, they are safe to use.