I was at my uncles house and I just watched his neighbor burn off his garden. He said that he would burn their garden every winter and told me that is better than just tilling your garden every summer.
I don't know anyone that does that. I do know some people burn their weeds.
The issue that I would have with burning your garden is that ash can be high in pH. So, if you had super acidic soil that could be something that could be beneficial, but I'd advise not planting things like blueberries or potatoes in that environment.
I think it makes most sense to pull certain plants (tomatoes, for instance, often harbor diseases that you don't want in the soil for the following year) and then try to compost the rest of the dead plants. Compost is going to be more beneficial than ash, overall, since it'll be chock full of nutrients which your next year's plants need.
There's always the option of just leaving the plants, as well, since it can help with run off, etc.
I don't till at all, and just layer compost, leaves, etc.
My neighbors on the other hand till every year and their garden is awesome, but because we live in a rainy area they start their garden much later than I do because tilling wet soil is going to do the exact opposite of what you want.
This year I'm experimenting with a cover crop of daikon radishes, so I'm pretty excited to see if that helps with soil structure and nutrients.