Burning those "Weeds" does little except release more pollutants into the air since the seeds that new "weeds" will grow from are already in or on your soil getting ready to germinate. Tilling those "weeds" in can be better than burning and tilling that plot several times, just as the "weed" seeds start to germinate, cna help some more. Many people have found that covering the soil with newspaper and covering the newspaper with a mulch material helps even more and with less work.

If you burn early, then you will get the weeds before the seeds are mature.
I agree that tilling annuals & a very few perennial will be better then burning.
With all other perennials, mow the ground & compost the clippings.
If you cut the weeds twice a mouth, in a year or two you should have all the perennials under control.
But you should talk to your extension agent before doing anything to weeds.
The agent knows more about your county, then all the out of state blogger on the net.
It will be your tax dollars at work for you.
Remember to take photo with you to his office.