The other answers give perfectly valid examples of e.g. a "grandfather's" clock, where you have to put in external energy from time to time - here, to lift the weight back up against gravity.
However, I feel that there is an additional aspect to the original question: The asker probably wants to stay completely passive and just harness the potential energy of the building, sitting on earth's surface, somehow.
To this, I say:
TL;DR: Theoretically possible, but practically meaningless (for now).
First of all, let's look at the basic formulae:
As the OP stated, a building exerts pressure on the ground. Pressure equals Force divided by Area:
$$P=\frac{F}{A}$$
So, to get the force a building exerts on the ground, we have to factor in Area.
No problem, the relevant area, the 'footprint' of the building certainly is known.
Now, what we want to get out of this is Energy. And Energy is Force times distance:
$$E=F\cdot s$$
And that's distance downwards (hereafter denoted by $h$), because that's the direction the force works and therefore that's where Energy is to be gained. The force is, of course, gravitational force, so we finally get:
$$E=F \cdot h = m \cdot g \cdot h$$
Now, we've got everything we need. Drop a skyscraper (let's say $m = 200 000 t = 200 000 000 kg$) down, say, $10m$, in a controlled fashion(+) and you're looking at
$$E = m \cdot g \cdot h = 200 000 000kg \cdot 9.81\frac{m}{s^2} \cdot 10m = 19 620 000 000J = 19.62GJ$$
that you could harvest.
However, as you might have suspected by now, this is a one-time-only-ever-fullstop-period-type of deal. You can get that energy exactly once (++) and then, never again. You have to put up at least the same $19,62GJ$ to lift the skyscraper back up and repeat the exercise, gaining nothing in the process.
And factor in the energy expenditure from digging the $10m$ hole in the first place, putting up some mechanism to drop the skyscraper in a controlled manner and harvest the energy, you're certainly looking at expenditures that greatly surpass anything that is to be gained.
So... it's not really worth trying.
Addendum:
'Futurologists' propose that a similar technique might actually become feasible in the far future. It has been suggested that a sufficiently far advanced civilization could use black holes to collect energy. The principle is very much the same:
You feed matter into a small- to medium-sized black hole - which is equivalent to "dropping stuff down". But other issues are going to be easier, relatively speaking, with this concept:
- You don't have to worry about any structural integrity of anything.
- Preparing things should be "easier"; no hole digging required.
- You can, mostly, just forget about the matter that has crossed the event horizon. You just have to make sure that the gravitational influence of the growing black hole doesn't get out of hand.
- The collection of energy is very much easier (again, relatively speaking): Matter getting pulled towards a black hole gets really, really hot due to tidal and friction effects; up to a point where a significant fraction of the mass-energy gets converted into radiation. Then, you just need to collect that radiation via solar panels or equivalent, probably arranged like a Dyson swarm.
There you go. Using gravitational potential is, as of now, only a theoretical issue, but might become feasible on a stellar scale in a few millions of years(+++).
(+) just throwing it down would probably cause it to collapse and release additional energy originally expended while building the skyscraper, but all that energy would be pretty much non-harvestable...
(++) at least within our earth's lifetime
(+++) man-made satellites do a similar thing right now with their gravity assist maneuvers, but that's probably not the scope of this question anymore.
