I've seen in multiple forms of fiction the sort of joking notion of stopping a volcano from erupting by just... plugging up the top with something? Like, a large and sufficiently heavy rock?

While the idea is obviously silly and extremely impractical, the prevalence of this trope got me thinking: is it actually impossible?

If you were to somehow fill in or "plug" the main crater of an active volcano, either with a single large object or maybe by filling it in with dirt or concrete or something, would that actually temporarily repress the eruption? Or would the pressure just blast the obstruction out again, or open a new vent out the side, or do something like Mt. Saint Helens did and blow out the whole side of the mountain? Just a fun thought, I know it probably wouldn't work but I'd wondered if anyone had actually done any sort of testing or research on the subject.

Yes... and no...

A volcano explodes (very simplistically, this is the problem) with a force. Apply the equation F=mA. We know the force of the volcanic explosion, we know the acceleration of Earth's gravity. Voilà! We know the mass of the rock we need to plug the eruption.

Sounds simple, doesn't it...

Outrageously ignoring the fact that volcanic eruptions occur for a number of reasons, we'll simplify the discussion to assume Mt. St. Helens is indicative of the complexities of the problem. And we can sum up a lot of that complication with one word...

Earthquakes

The eruption of Mt. St. Helens was preceded by a number of earthquakes of various sizes, ranging from itty-bitty to magnitude 5.1. Earthquakes have the nasty habit of softening up and shifting around rocks, faults, vents, and everything else associated with volcanism.

But the fact of earthquakes isn't all, where they take place is also a problem. Some are deep in the earth, others are shallow. Some are miles away, some are right under the honking volcano. And they all add up.

And we're not done

Just to make things ugly (remember, I'm already simplifying by ignoring types of volcanism), you have the geological makeup of the surrounding area. Do you have a really tall mountain with thin walls? Or a really low one with thick walls? Do you have a big throat or a small one? Is the Earth desperate to pop this proverbial zit? Or is it just trying to clear a little mucus?

Conclusion

Not to put too fine a point on it, but you're trying to fix the Earth's "grumbly tummy" by putting a billiard ball in it's mouth. The billiard ball is certainly more mass than whatever might come out of the stomach,¹ but if you tried it yourself, you'd likely be surprised to discover that it didn't stop much — in fact, it usually makes it a lot worse.

So, yes, for some (few) eruptions dropping a big ol' rock on top will stop it. For the rest... you'd better have evacuated the area, first.

But it'd sure look cool, wouldn't it? Kinda like when they tried to get rid of the beached whale by blowing it up.

Oh, and just a reminder, musket rifles use the same basic premise of putting a billiard ball atop a grumbly tummy. And what goes up....

_{¹ In my metaphor. It's my metaphor, after all.}

Every volcano that isn't actively erupting is actually just like this

All volcanoes that are not erupting are currently plugged up by something. Usually it's rock or cooled magma. The better the volcano is plugged the more pressure and heat it will need to erupt. It's also possible for it to never erupt, but then you might say that it really isn't a volcano anymore, because, after all, if you dig deep enough anywhere on the earth you can get down to molten rock, and then you have a volcano. The volcano under your house is very well plugged right now (please don't start digging).

No

You cannot plug a volcano.

Volcanoes erupt because:

1. The magma is very fluid, and there is a new influx of magma coming from the mantle. This magma is very efficient in carving a way through solid rock at extreme pressures. When it gets close to the surface it will just make make new dykes and sills to flow through. These type of magmas will be very fluid, something like Hawaiian magmas. Your attempt to plug it would be like putting a piece of paper in front of a bulldozer.


2. The magma reached gas saturation, and it is rapidly nucleating bubbles of gas (mostly H2O), and it's going to explode. Your plug might only make it worse, because you're blocking the escape path of the gas, allowing overpressure to build up. Plugging it only delays the inevitable. Again, it's like putting a small piece of paper on top of a mentos-coke combination, hoping that it will stop it exploding. It will not.

Submarine Volcanoes Suggest NO

You are asking whether we can plug volcanoes by putting something heavy on them. Well, consider the volcanoes that erupt at the bottom of the ocean:
https://en.m.wikipedia.org/wiki/Submarine_volcano
There are tons of these volcanoes, and many sit at depths where massive weight of water is pushing down on them. That’s a pretty effective plug. But they still erupt regularly, and some throw magma quite a distance vertically. The pressure below is more that the oceanic pressure. You can get some pretty freakin’ heavy rocks and plug specific volcanoes for a time, but the pressure to erupt is going to win eventually.