[Mithadan]

If I recall correctly, the Lost Road (HoME 4) may suggest that the final destruction of Beleriand came during the fall of Numenor. However, as is the case with much of JRRT's earlier materials, this may not be a final conception.

My belief is that Beleriand was destroyed gradually during the 50 years of the War by the the host of the West against Morgoth. Some portions of Beleriand may have survived the war only to gradually subside. This would have allowed the remaining elves and men who dwelt there to escape.

As to why it was destroyed, Morgoth, in an attempt to slow the Host of the West, may have expended much of his remaining power by opening cracks in the ground, causing avalanches and earthquakes, etc. The Host of the West may have also been involved in undermining Beleriand's structure by rooting out strongholds which Morgoth may have established in the various mountain ranges. At least one person I know claims there is textual support for the idea that Morgoth created a series of tunnels under Beleriand to assist in the movement of his forces.

I advance one final, somewhat radical, idea. When all was said and done and Morgoth was defeated, what remained of Beleriand may have been so severely tainted and damaged that it was beyond redemption and the Valar themselves may have sunk it after ensuring that all men and elves were safe.