A saw is the tool of choice for removing a burl from the side of a tree. I'd say use a chainsaw for efficiency, especially on large burls. They can grow to several feet wide.
Drying the wood is one step I didn't see mentioned. It would be something you'd have to consider though.
Wood works more easily when it is green, wet, and fresh off the tree. Its moisture content fresh off the tree can be in the 50% or greater range. NW Coast Native wood sculptors/carvers sometimes drape wetted towels over their "in progress" totem poles overnight to keep the wood damp enough to carve/sculpt easily the next day.
Typical moisture content of wood furniture & products inside your house is likely in the 8-12% range. That is a considerable loss in moisture (and weight) from the green moisture content level. That moisture loss as it dries causes wood to shrink. You need to accomodate that movement to prevent checks, cracks, splitting, and buckling. Burls, being typically irregularly shaped make it harder to achieve an even drying rate for the whole of the piece to prevent splits, compared to the even surfaces of rectangular milled lumber.
As noted, gouges and chisels will work the burl into a bowl.
As for the survival of the tree, a contiguous filament of cambium from the roots to the leaves seems to be one of the minimal requirements. So unless removing the burl from the tree somehow circumscribed a ring of cambium from all the way around the tree, the tree would likely survive. However, opening the bark offers an entry point for insects, so tree wax would be in order to seal it up.
In Japan they venerate old trees and prop up the branches of some ancient ones with crutch-like supports to keep the sometimes half-rotted base of the tree from snapping off. It seems that as long as there was a thread of cambium (and sometimes it was very very narrow, like an inch or so wide is all) making a food path from the roots to the branches, it would keep the upper part of the tree viable.
If you go to Google images, you can find pics of bristlecone pine trees that also illustrate this. Much of the trunk of the tree may be bare wood. But there is a strip of bark connecting the roots to the branches.
