I have wondered this awhile. In the Seven Harrys chapter, it seems that after being foiled at Privet Drive, the Death Eaters and Voldemort would have immediately thought of -- and then gone to -- the Weasley's house at the Burrow, even that very night. It seems odd for the DE's to not have thought of this, or at least have tried. Then they could have sent a huge force of Death Eaters and allies against the Order and Harry and ended it right there. Is this a plot hole, or am I missing something?
In addition to Richard's correct answer, there is this observation by Snape:
'Where are they going to hide the boy next?'
'At the home of one of the Order,' said Snape. 'The place, according to the source, has been given every protection that the Order and Ministry together could provide. I think that there is little chance of taking him once he is there, my Lord, unless, of course, the Ministry has fallen before next Saturday, which might give us the opportunity to discover and undo enough of the enchantments to break through the rest.'
Harry Potter and the Deathly Hallows - p.12 - Bloomsbury - Chapter 1, The Dark Lord Ascending
And, indeed, moments after the Ministry fell, Death Eaters did descend on The Burrow. Kingsley had just enough time to fire off the Patronus. After it arrived the dust had barely settled before the Death Eaters were there.
In short, then, the answer is that the safe houses were extremely well protected by both the Ministry and the Order. Even the Death Eaters couldn't blast their way through the enchantments without inside knowledge gained from the Ministry.
Also, you forget, Voldemort operates by stealth. Even when he has taken over, the coup is as silent as possible and, officially, Thicknesse replaces Scrimgeor. He is a schemer, Voldemort. He doesn't just attack a high-profile Ministry target all guns blazing. He infiltrates the Ministry like poison, subdues it and then mounts his attack.
Voldemort didn't know which of the (many) safe houses Harry was heading to. As an added precaution, he then used a port-key to jump from Tonks' parent's house to the Burrow.
Given that each location was equally plausible and given the risk in moving against the Order of the Phoenix prematurely, Voldemort seems to have chosen not to chance an attack against each of the likely sites, relying instead on catching Harry during his move
‘The one thing we’ve got on our side is that You-Know-Who doesn’t know we’re moving you tonight. We’ve leaked a fake trail to the Ministry: they think you’re not leaving until the thirtieth. However, this is You-Know-Who we’re dealing with, so we can’t just rely on him getting the date wrong; he’s bound to have a couple of Death Eaters patrolling the skies in this general area, just in case. So, we’ve given a dozen different houses every protection we can throw at them. They all look like they could be the place we’re going to hide you, they’ve all got some connection with the Order: my house, Kingsley’s place, Molly’s Auntie Muriel’s – you get the idea.’
I am surprised nobody here mentioned the very important detail that Harry did not head to the Burrow at all from Privet Drive. The 7 Harrys head off in different directions with the original Harry (& Hagrid) heading to Nymphadora’s parents house. From there, Hagrid and he take a Portkey to the Burrow. So Voldemort did chase Harry all the way - but to Nymphadora’s parents house, not to the Burrow. After they crash-land at the Tonks’, here’s the excerpt from the book:
"Now Harry understood why Voldemort had vanished; it had been at the point when the motorbike crossed the barrier of the Order’s charms. He only hoped they would continue to work: He imagined Voldemort, a hundred yards above them as they spoke, looking for a way to penetrate what Harry visualized as a great transparent bubble.”
There was no way Voldemort could’ve known (based on what Snape had revealed) what the plan was after getting to the Tonks’. So, the Burrow doesn’t enter the picture at all.
Firstly, they actually went to Tonk’s parent’s house and used a port key. Second, they didn’t know which house it would be. Third, the order may have used say, the Fidellius charm.