Yes, but very little
Hard disk platters are mainly aluminum alloy, consisting of one or more aluminum or glass discs, coated with magnetic materials. Because of its low type variability at high temperature. The main aluminum alloy based brands are Seagate, Maxtor, Samsung, Western Digital and Easytec.
The head part is made of alloy and permanent magnet.
The shell is basically made of lightweight aluminum alloy.
The circuit board is relatively small in the entire hard disk, and the data cable and power cord jack are coated with copper metal or gold.
In summary, hard drives contain almost no precious metals.