VOC and drying speed
In general, clear coatings today have been reformulated to product much lower levels of VOCs (volatile organic compounds) then even three years ago. Not knowing what product he plans on using, varnish, urethane, lacquer and the age of the product makes it difficult to guess how long the finish is going to take to gas off.  Most modern products gas off in about 24 to 48 hours. Although you may still smell a slight "new paint" smell, most of the irritating fumes are pretty much gone after a day or two.  Just ask your landlord to give the doors a few days to cure, not dry, before installing them.