Some people react to even tiny traces of varnishes and sealing polishes used on solid wood furniture. Furniture that is not new is usually little problem. Furniture made before the 1940s is usually trouble free, since many varnishes, lacquers and polishes used then were water-based, so buy or use older furniture for preference.
If you are exceptionally sensitive to varnish fumes, or need to buy or have made a new piece of furniture, you could use water-based varnishes rather than solvent-based ones. These are well tolerated, but do not give full protection against splashes, spills or marks, and are thus not really satisfactory in everyday use. There are alternative compromise solutions which are satisfactory. If you have furniture made specially, ask the maker to use a varnish of your choice. If you buy new wooden furniture, allow it to air off varnish before use.
French polish, shellac and Japanese lacquer are solvent-based and can cause sensitivity when being applied, and shortly after, while vapours are being released. Do not use these if possible. If you do use them, ventilate well afterwards and leave the piece of furniture to air before use. If using adhesives in upholstery or repairs to furniture. If you are sensitive to resinous woods, like pine and cedar, look for furniture of less troublesome wood, such as beech, ash or oak. It may help to apply several coats of varnish to pine or cedar furniture to cut down fumes. Varnish the inside of drawers and cupboards and shelves as well. If you are sensitive to enamel paints on metal furniture, sniff carefully before buying. Wash down surfaces with a solution of one dessertspoonful of domestic Borax in a bowl of warm water. Allow to air before use and keep away from sources of heat.
Use glass and marble furniture, and mirrors, stone, slate or ceramic tiles and surfaces if you can.