Why does my clothes dryer make a rattling sound after 5 minutes?
A dryer is a very simple machine. 

There is a belt that turns the drum and there are bearings on the drum supports.The motor, the belt, or the drum bearings can have a problem - the drum bearings is what I suspect.  

Sometimes the simple things are overlooked so with that said:

Another possibility is that you have something on the side of the dryer (like a dryer shoe rack) that starts to rattle against the machine after it is spinning up a bit, also check your venting pipes.