your priority is to hit dh when he tries to hit you. If he tries to cancel your burrow with dh you have pretty much won. If he sets up 2 - 3 aows you better off cancelling the rush after they have finished.
I don't recommend sending so many peons at first (wait and see how the burrowrush turns out!). I think three peons is a good number because you get the burrow full and 4 peons are enough to repair it for the time being. If you take so many peons with you AND sell your tp your tech won't be any faster or even worse (since your base is lacking wood peons and you don't have any quick means of transportation), slower than your opponent's.
Specifically for the map EI, the NE would have just crept the Ogre/Troll camp if he AoW crept, and if he was Hunt creeping, the AoW would still be under construction. For both cases your hit would be timed when the NE is relatively weak and still re-adjusting from his opening. Granted, the NE will have an item advantage on your BM, you have the advantage of having multiple high-priority targets. With proper micro like capu mentions (only hit the DH when he hits you, don't go chasing after him), you'll minimize any advantage he might have.
If the NE counters by placing additional AoW's up, then your best likely reaction is to allow them to go up and simply abandon the rush. The key here is that you are the aggressor, and he is the defender; if the NE were the one pushing you, then taking out AoW's would be a priority. However, as you are the one doing the pushing, you have the option of abandoning your current position.
Things you should keep in mind when you see those AoW's go up are:
Has the NE canceled his tech?
How many AoW's has he put up in response?
How many Archers has he pumped in response?
How dry are his Moon Wells?
Have your Burrows finished constructing?
How do you match up to him in food count/resources?
The big factors are his tech, how many AoW's he's put up, and if your Burrow(s) have already finished construction.
Observing his tech allows you to see how committed he is to stopping the rush. If he's chosen to stay teching, then he shouldn't be able to put up too many AoW's and any you see going up will delay his tier2 buildings; you should be able to punish him severely for playing greedy by staying and committing to the rush. However, you should tech yourself when you get the chance to keep yourself even. If he's chosen to cancel his tech, then you can expect him to over-commit resources in order to fend off the rush. Stop reinforcing the rush and break it off while teching to gain an easy lead that you can handily ride to a win at tier2.
Observing his AoW count allows you to see just how far ahead you are. In most cases, he won't have the resources to tech now that he's committed himself to possibly multiple AoW's. Your best course of action is to break off the rush and ride the tech advantage. The NE also shouldn't have enough resources to produce from all his AoW's right away.
If your Burrows have already finished construction, chances are it's best to stay the course, especially if the AoW's are still under construction.
credit : d1ce