Obsidian is glass and therefore is cut exactly like glass.
While there are lots of methods of stress and thermal breaking of thin-walled glass, I get the feeling that everyone is imagining a block or slab of obsidian. You’re not going to like the answer then.
For large pieces you have two general choices: grind it down or melt it down for molds or extrudsions.
To answer your question grinders tend to be steel (or alloys), carborundum, or diamond blades. There are more exotic materials, but mostly that’s what you get.
MKDesign 