What is the long term impact of large scale removal of material from subterraneal caches on the structural integrity of the soil?
The answer to your question will depend on the type of material and the depth. I will restrict this answer to groundwater withdrawal to focus the discussion.  Of course there can be effects from other extractive industries. 

Groundwater pumping can lead to large scale subsidence, with the Central Valley of California being one of the best described examples. According to that website, "More than 80 percent of the identified subsidence in the United States is a consequence of human impact on subsurface water."



Overall, the extreme subsidence shown may cause damage to foundations, etc. but is spread over a large area and is pretty even, unlike your classic sink-hole.  It is the differential settling that causes the most damage.

There are several types of sink holes (also called dolines), they may be related to failure of indurated rock above cavities or by failure of soft soils/sediments. The most common natural sinks occur through the dissolution of limestone in 'karst' terrain and they can be kilometers across. This can lead to the creation of water or air filled caves. In some cases a cavity forms in sediments above the limestone and can subsequently collapse. Cavities can also be formed in sediments where they are removed through pipe erosion but that tends to be a shallow phenomenon (but can still be damaging).

The description, above, talks about natural processes but these can be enhanced by groundwater pumping.  The water supports some of the overburden and removing it weakens the structure and can lead to collapse. Flow systems can change, enhancing subsurface soil erosion as water tables decline. And of course, building over a cavity may exceed the load bearing capacity.

So the long term impacts will depend on both the geologic setting and the severity of water level decline. Will this have a major effect on global infrastructure and economies? I suspect not but the damage we have seen in certain areas is likely to continue and perhaps spread.

A somewhat old but good reference on the subject sinkholes and caves that discusses the land-use issues is Geomorphology and Hydrology of Karst Terrains. White, William B. (1988). Oxford University Press.