When wasteful war shall statues overturn (5)
This war is certainly violent, but it's less about killing people than about destroying art. The speaker is imagining a cultural calamity that will one day wreck all the beautiful and meaningful things that humans have created.
And broils root out the work of masonry (6)
This line repeats the same idea of line 5, but with different images. Here we get "broils," a wacko synonym for war that sounds like a cross between a pimple and a steak. And instead of "statues" (5), we have masonry, which refers more generally to stuff carved out of stone. But the message about artistic devastation remains the same.
Nor Mars his sword nor war's quick fire shall burnThe living record of your memory (7-8)
Here the speaker switches from talking about war will do, in all its gory detail, to what it can't do. The point is to rev us up about his poetry and its awesomeness. Yeah, so war destroys pretty much everything we've created out of actual material, but it can't touch something that has no substance, like a memory. Or a poem.