Is a lich aware of its phylactery being destroyed?

Possibly immediately, but at the very least he will figure it out sooner or later, how sooner or later is up to the DM.

The Monster Manual entry for liches says:

A lich must periodically feed souls to its phylactery to sustain the magic preserving its body and consciousness. [...] A lich that fails or forgets to maintain its body with sacrificed souls begins to physically fall apart, and might eventually become a demilich.

So at the very least a lich will begin to notice the degradation of his corporeal form.

There is evidence, however, that a lich may immediately know that its phylactery has been destroyed. In Waterdeep: Dungeon of the Mad Mage, the party can encounter a lich; we see this statement about it:

If the characters destroyed [the lich's] phylactery on [spoilers], [the lich] attacks them on sight. [The lich] has no beef with them otherwise.

It is not spelled out how the lich might know the party destroyed the phylactery, but the lich seems to know it was destroyed when the party encounters it.

Further details, W:DotMM spoilers ahead.