We had concerns over affects to our pets. As I understand, the chemical used with tenting basically displaces the oxygen and the termites sufficate. You should talk to the company doing the work directly to get a understanding of the process and any risks.
How harmful is the tenting to our health? Please be honest
I have had no experience with 'tenting' but its likely that you are caught between a rock and hard place.
If it isn't tented, eventually the termites will eat your place down and you will have to find new accommodations.
If it is tented, there may be a possibility that you will have to move because it affects your asthma.
The house is being "tented" so that a fumigation can be performed. They will use a gas that will penetrate the structure over a period of time. The house will be ventilated after the proper time. Fumigations leave no residual inside or outside the structure. Should be no medical concerns. As vsheets said, you should take your concerns to the company as they do not want problems either. Also talk to your medical pros when you find out what product they will be using.