Is it ok to have bonfires on my garden bed when I'm not growing anything? Should I do anything special with the aftermath or just rake it into the soil?
I wouldn't do it on my garden bed:
I'm not a fan of bonfires in general:
You can make an exception for diseased plant material -- you don't want to spread disease around by composting it. When I want to get rid of diseased material I will put it on top of a hot fire in my wood boiler; understandably not everyone has a wood boiler available.
If you decide on a bonfire, I'd suggest doing it somewhere not on the garden bed. The ash left over from burning wood will have high pH and will contain some potash and calcium. You could spread the ash on your garden bed if you want to raise the pH of your soil.
Never burn pressure-treated wood or manufactured wood (e.g. OSB, plywood).