English equivalent for the Persian proverb "The mountain just gave birth to a mouse"

full of hot air refers to someone who talks a lot and makes big claims, but with no action behind their words.

Similarly there's all talk no action.

The phrases essentially mean the same thing, the first more emphasises the boastfulness of the subject, the second more emphasises the lack of action.

These terms can be used to refer to anyone - not just people in positions of authority or expected influence.

That politician is full of hot air, he makes all these claims about how he's going to make the world a better place, but really he's all talk and no action.

This is a famous proverb in virtually all languages. Best known perhaps from the Roman poet Horace (1st century AD): "Parturient montes, nascetur ridiculus mus" (The mountains will be in labor, and a ridiculous mouse will be brought forth). A common English version is "A mountain gave birth to a mouse".

The term damp squib comes to mind.

A situation or event which is much less impressive than expected: my moment of power was a damp squib

_OED

So, in your intended context, you could describe either the under-performing politician or the unimpressive event as a damp squib:

He said he was going to start a revolution but it turned out to be a bit of a damp squib

The Free Dictionary gives a nice note about the origin of this term:

A squib is a type of firework (a small container filled with chemicals which explodes to produce bright lights and loud noises) and if it becomes wet, it will not explode.

The incendiary root dovetails nicely with your idiom's volcano metaphor.

All talk and no trousers — TFD

Full of boastful, arrogant, or shallow talk that never materializes into results. A variant of "all mouth and trousers," meaning the same thing. Primarily heard in UK.

"The team's manager keeps promising title after title, but he's seeming like all talk and no trousers at this point."

There are different versions of this idiom — "All something and no some other thing"