Work only in well-ventilated spaces, such as the yard or an open garage, and dispose of unused varnish after you finish if you do the work yourself. Insist on a guarantee of low-VOC varnish if you take the furniture to a professional refinisher. Most are happy to discuss their methods and materials.
Place the furniture in a well-ventilated area such as the garage or a garden shed for up to a week before bringing it in the house.
Ventilate your house once you bring the piece in. Open windows, run bathroom and kitchen fans and use a whole house exhaust fan if you have one.
Change furnace and air filters after you cannot smell the solvent-smell of VOC off-gassing anymore. Heat and humidity can revive VOCs that adhere to filter media.