It do NOT need to be made from precious metal. But usually, most feel that precious metal is most fitting for a parish that can afford it.
Regardless, the container/casing should be made of metal & glass.
For example, the local parish near me has a wood monstrance, carved in the image of Mary for use in their Mary Chapel. The Eucharist appears as her womb. However, the casing for the Eucharist is still metal & glass
They have another, more typical metal monstrance for the main altar and sometimes used in the Mary Chapel too.