A CPAP does not require mouth breathing. Nasal pillows require you to breathe through your nose. If necessary they shut your mouth with a chin strap. A piece of 3m tape sounds dangerous for someone if the problem is nasal obstruction or rhinitis rather than their mouth dangling open. They might not be able to gasp for breath and experience an even worse apnea.
I read Sleep by Nick Littlehales and he mentions that taping your mouth closed is one technique you can use - I think I recall he says some British athletes (cyclists?) use it - but doesn't recommend it for anyone with a medical problem.
A CPAP does not require mouth breathing. Nasal pillows require you to breathe through your nose. If necessary they shut your mouth with a chin strap. A piece of 3m tape sounds dangerous for someone if the problem is nasal obstruction or rhinitis rather than their mouth dangling open. They might not be able to gasp for breath and experience an even worse apnea.