Ceramic kilns fire at very high temperatures and the clays and glazes inside can contain a wide range of materials that produce fumes when heated.
Fumes can be produced when a clay or glaze decomposes during heat-up in the kiln. Fuel burning kilns produce hydrocarbons as a product of the fuel they employ. However, since kilns fire to temperatures sufficient to actually melt rocks, they can release vapors generated as materials in the glazes and clays decompose. The fumes are akin to a volcano (albeit on a much smaller scale). In extreme cases of poor ventilation vapors can deposit as very tiny particles on walls and other surfaces. The individual particles are so small that they can be inhaled deep into the alveoli (air sacs) of your lungs.
The most common fume associated with kiln firing, other than CO2 and CO, is sulfur and it is easy to identify from the odor. Many clays contain it (in the form of calcium and magnesium sulphate), as do fuels. Organics used in decoration (e.g. wax) produce fumes as they burn. Some metal oxides volatilize and form fumes (notably copper, zinc, manganese), these can lead to metal fume fevers of different types. Manganese in particular can be quite toxic.
Is is common sense to arrange for some form of ventilation for your kiln to avoid breathing these fumes. All kilns sold today have guidance on doing this.
Are neighbours paranoid about your kiln filling the air with silica dust? Silica is not volatile and does not produce fumes from a kiln. The silica is reactive during firing, busy forming silicates in the glaze and clay body. Silica and silicates make up more than 60% on the earths crust. Your neighbour is exposed to more silica dust when taking a walk, sweeping a floor or gardening than from the stack of your kiln. Gas kilns put out carbon and sulfur compounds, like a car, when fired. These come from the fuel, but also from carbon and sulfates in the clays and glaze materials, as these decompose during firing. Potters make rocks. Rocks are stable, they endure millions of years. If your neighbours want totally clean air they need to move to a place where there are no cars.