The DNA double helix is held together via hydrogen bonding between base pairs. If one was to separate the double helix it will want to reform the double helix to minimize hydrogen bonding potential within the background hydrogen bonding potential of the water. A triple helix would contain too much residual hydrogen bonding potential. This would cause nonequilbrium for all the enzymes used for genetic expression. These enzymes would not want to approach and bind to the DNA. Instead the DNA would attract protein materials needed to help lower its hydrogen bonding potential.
Theoretically, one could modify the unpacking and transcription enzymes so that they are in equilibrium with this higher hydrogen bonding potential state of the DNA. But this would alter the hydrogen bonding gradient to the cellular membrane and cause the cellular gradient (positioning of materials) to shift, requiring new proteins to maintain the gradient. This new protein train may not be able to provide for the cellular diversity needed to express life.