Do divergent tectonic plates destroy mountains?
Quite the opposite. Divergent boundaries can cause mountain (well, actually volcano) building, because upwelling magma is part of the rifting process. 

First and most importantly, the Mid-Ocean Ridge can be considered the longest, most massive mountain range in the world. 

It's more obvious, of course, 
 on continental rifts like the East African Rift. Mt. Kilimanjaro, a product of rift volcanism, is the highest mountain in Africa. 

Another famous locale of volcanism at a divergent boundary is Iceland, although a hot spot may play a role there. 



Source: USGS (via usu.edu)

But BUT BUT it seems you want to know what happens to existing mountains. Some others have answered that only erosion destroys mountains, and that's mostly right, but only because it's a matter of timing.

It's often difficult for us humans to comprehend just how big geological time scales are. Most of the time, a mountain range created by a continental collision is eroded away completely by the time the next rift comes along. 

This was the case along the eastern seaboard of the United States. The mountains created by the collosion of Africa and North America in the Carboniferous (the Alleghenian Orogeny) were almost completely gone by the Triassic, when the Atlantic Ocean began to form from a rift along about the same line. 

However, we have an example that might result in a qualified "yes" to your question, a little closer to the present. This is the way the Yellowstone Hotspot moved across what would become the northwestern United States.  One of the things that caused geologists to recognize the Yellowstone Caldera was the 80km gap it melted in a mountain range it passed under, separating the Gallatin Range in the north from the Red Range in the south. 

The hotspot's track might just be the genesis of a new rift that breaks North America apart, given enough time and the right stresses. 



(Source: USGS)