Fuel furnace gives off fumes every time it ignites, is this normal?

Strange smells? Smoke? What are we talking about when you use the word Fumes?

Websters defines fumes as: 1 a: a smoke, vapor, or gas especially when irritating or offensive <engine exhaust fumes> b: an often noxious suspension of particles in a gas (as air)

I'll hit on several points since I do not know exactly what you mean here.

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Fuel consuming furnaces are vented to the outside. Electric furnaces need to vents.

If the vent is blocked then the gases of combustion stay in the house, this is not a good thing.

Carbon monoxide is one of the many by-products of burning which is deadly.

Fuel (Liquid fuel) or "oil" furnaces should put off a near invisible smoke, if anything is seen coming out of the stack it should be white, meaning water vapor - especially when the warm moist air hits cool winter air. Much like the white "smoke" one sees from car exhausts on cold days.

Any "smoking" means a poor air to fuel ratio or "mix". Some, not all, liquid fuel burning furnaces were not build to handle "heavy" exhaust - or partially burnt fuel, thus it settles out going down the flue, not up. This is still not a good thing. Either fixing the mixture (air to fuel ratio) or adding extension to the stack to produce more draw.

Fireplace chimneys (As a good example) have to be taller and have a larger top opening to fire box opening to provide the correct upward draft. Many need to be pre-heated before they get a fast enough-upward draw.

BTW Chimney design is not an exact science. There are no fast and hard rules, oft time different things are "tried" such as insulating the interior chamber of a flue so it will retain heat and cause the gasses to rise faster, or to extend the flue catching crossing breezes which "suck" air up.

Assuming for a moment that your fuel is not burning properly and that once the furnace has been on for a while the "fumes" dissipate, then this can be your underlying issue. This mean calling a chimney person to come out and extend your stack a bit.

Perfectly good chimneys can be affected by new building where the air currents/wind patters change slightly upsetting the draw of the flue.

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Smell of something burning, other than fuel:

Dust accumulates on the burners of any and all furnaces, stoves, heating elements, etc. Dust and debris can build up in ducts, on the inside of flues, in the combustion chamber - everywhere that normal house cleaning can not possibly reach. Heating the area will result in a stench.

Things like small animals can get trapped in places. Imagine a dead mouse between the top covering of ones furnace and the combustion chamber - the smell of "burnt" meat and smoke.

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Duct work: Duct work becomes pretty foul pretty fast, dust, debris, dead critters (spiders, ants, mice???) mold, mildew all collect in ductwork. If the ductwork is duel purpose for heating and Air Conditioning the moisture of summer from the A/c condensation can grow many interesting things in the ducts. Come the heat of winter the "burn off" or the heating causes interesting foul smells.