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The effects of any nearby mass would be negligible for all practical purposes. In any event, a plumb bob and spirit level would experience the same forces and should agree. The effect of the gravitational pull of the sun and moon would be greater as evidenced by ocean tides. Any edifice constructed near a very large mass such as a mountain would be plumb and level to the eye but in actuality, it would be slightly askew. There would be no simple way to check this as any reference would see the same effect. One possible way would be by observing the position of astronomical features although this would require very precise time measurements. An interesting thought.Yup - this happens! My Dad (builder) had a traditional lead plumb bob set into a "teardrop shaped" cutout in a hardwood plank, and with a fine V-line cut into it. The whole thing, which stood as high as a man, had a cover box that could be removed sideways, with viewing aperture, all to stop air disturbances. He told me the same thing about Table Mountain (Cape Town). The middle of that peninsula mountain is made of iron, or something dense, because all the buildings stand slightly off true geographic vertical, and plumb bobs hang slightly pulled toward the mountain.
Not a problem for building purposes, because the forces in the buildings are still properly catered for by regular spirit levels. I expect any satellite tracking dishes in the region would have to correct for clinometers being somewhat untruthful. The messed up way levels worked in Cape Town has been well known since back in the 19th Century because of the importance of the Observatory in Cape town, and the huge efforts to establish the Southern Arc of the Meridian, and prove the Southern half of the Earth oblate spheroid was the same shape as the Northern half.
The deviations from vertical were known as "La Caille’s discordances". I am not sure it is true that at the base of the mountain, coffee in a cup will look slightly skewed off level. I never noticed anything like that, and I lived for a while in a University residence on the slopes of Table Mountain.
View attachment 386582
Great history and pictures here -> Arc of the Meridian Survey
There has been some locations where whole mountains have been leveled due to mining operations. Any edifice would now be leaning slightly away from the missing mountain. I suspect that the change would be insignificant. Certainly, we wouldn't be able to observe the effect as our internal plumb bob that allows us to observe the vertical would see the same forces.
An interesting experiment would be to level a stable surface with a sensitive level and obser4ve any changes over time. These changes, if any, could be compared with the position of the moon and the sun over the course of a month to see if there was a correlation.