Can I die from de Quervain's tenosynovitis if I ignore it?

NO you won't die! But you will suffer with pain if you wait. This is a condition easily treated and you will be grateful for it! get a cortisone shot if you haven't had it already. Surgery if shot doesn't work!
Not likely. This is an inflammation or irritation not an infection of the tendon. It can be painful but it is not likely to cause death. Rest and anti inflammatories like advil (ibuprofen) (if you can take them) usually helps. Sometimes a splint may also be necessary.