Invisalign will not interfere with breathing, so if you're worried about having asthma or stuffy nose from common allergies Invisalign will not make breathing more difficult. There have been rare cases of allergy to the material used in Invisalign so if you're worried about that you should visit your orthodontist and ask about it. If you have known allergies to any form of plastic you might want to be tested for the type of plastic used in Invisalign before spending the money to get treated with it. Just for comparison, allergies to metal and latex which are often used in conventional braces are probably more common than allergies to the plastic used in Invisalign.
You should not have any problems with the Invisalign aligners. Breathing should not be a problem because they are very thin. The material is a type of acrylic and is fully cured when they arrive as a finished product. Talk to your dentist if you have specific concerns.