The blackTooth is a laser cutter/engraver, and its current design directs the beam in 2 dimensions (X and Y) over the surface of a workpiece which is assumed to be lying flat beneath the laser head. There is no mechanism to move the Z-axis directly. There is likewise no direct provision to move either the workpiece or the laser head for a 4th or 5th axis of operation because that would change the optical path and/or the focal point. However, you could certainly add a 4th axis to move a workpiece beneath the laser head -- it would be an interesting project, and if you do it we would like to see pictures.
The ability of the software to support this addition depends upon which software you are using. If you are using a breakout board and machine control software designed for conventional CNC mills, such as Mach3, LinuxCNC, or PlanetCNC, then the answer is yes, the software will support the 4th axis quite easily. If you are using the integrated laser controller, the adaptation would be more difficult.
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Does your hardware & software support adding a 4th and 5th axis to the Black Tooth?