A cheap GPS device can give you millimeter-level measurements every few hours, so using an active satellite does seem like overkill.
What you are possibly referring to is called dimensional control in geomatics and civil engineering, and there are various measurement techniques that can be applied, including GNSS-based.
In a way, the hardest part of it is the planning phase, where you have to decide which technique is most appropriate to meet precision, accuracy, timeliness, and cost constraints. All those factors kind of fight against each other. Finding the sweet spot is non-trivial.
Edit: And worst-case, you should be able to do an exact survey once, and then you can remember that and know which phase to use in the future. (Barring sudden multi-cm shifts, but in that case you have bigger problems.) You still have atmospheric fluctuations to deal with, but that's why you're measuring all day.
https://news.ycombinator.com/item?id=13040765
The quoted horizontal standard errors of the daily position measurements are ~1.5mm. The vertical standard errors are ~5-7mm. (Not as bad as a factor of 10, but certainly worse than 2x.)
The vertical position measurement responds well to averaging of daily errors to beat down the RF propagation effects that cause them.
If you're at the top of a skyscraper you can mitigate this somewhat by looking at satellites closer to the horizon as well -- this solves the geometry problem -- but then you run into increased noise from the larger amount of atmosphere you're looking through.
I Googled and found some vague and unsatisfying explanations, but, luckily, wikipedia tells us how to do the math: https://en.wikipedia.org/wiki/Dilution_of_precision_(navigat...
The post processing technology was pioneered by NASA among others. In the western US there is a network of over 1000 such monitors, anchored very deeply, that use these methods. Here is the time series from one in the Bay Area: http://www.unavco.org/instrumentation/networks/status/pbo/ov...
In that plot, you can see lots of interesting effects. First, the virtually continuous lat/lon velocity. Second, the annual vertical trend which usually has to do with groundwater. Additionally, smaller and larger perturbations from all kinds of sources from nearby construction to large scale seismic effects like distant earthquakes.
GPS of this kind is a complementary technology to the fantastic InSAR measurements featured in TFA. Both are widely used in geodesy to measure seismic deformations and deformations due to groundwater extraction. The two approaches offer different temporal, spatial, and accuracy trade offs. For a building like this, if you really wanted to measure subsidence, GPS would yield more temporal information than InSAR.
If I recall correctly, the fundamental limit to the GPS accuracy is uncertainty on tropospheric propagation of the radio signal.
I don't know if this technology is readily available for one-off commercial applications, but it wouldn't be surprising. The required accuracy is clearly within reach, and the software to do the post processing is widely licensed (https://gipsy-oasis.jpl.nasa.gov).
The vertical precision of GPS is quite low, so measuring subsidence with gps is very difficult.
The first few paragraphs are about the Millennium Tower, but that's just the hook. The rest of the article discusses the wide-area applications, including SF, elsewhere in the Bay Area, and Oslo, Norway.