where are metal ore mines that are closed down or that their activity
is diminished considerably because the ore deposits have been
exhausted?

Everywhere that there are or were metal mines, I would think.

Around the old gold towns in SW West Australia there are many abandoned 
gold mines, also in the western US states, tin mines in England, silver 
mines in Europe, ...

I suspect nearly every state in the USA has at least one abandoned
metal mine ie Goldhead State Park in Florida, about 45 miles South of
Jacksonville, Florida.   Neil

Plate Techtonics creates certain zones containing more metals than other zones,

Some old mines are reworked when Technology Allows easier extraction
of lower quality ores.

Major strikes of good quality ores can close other mines overnight.

Some mines never had any ore and were used to milk investors.

Some old mines get a new life storing record or radioactive wastes.