When starting a radial engine, smoke is inevitable, plus radial have a tendency to seep oil. They say if a radial isn't leaving drips on the ground, it's out of oil. Once running though, they should run clean except when running a rich mixture (typical when breaking in an overhauled engine). Inline engines
may smoke on startup, but typically do not.
Here is what you get when you start a radial:
If it's had too much prime, or if you have spun the engine too long with a startup, it can result in this:
Now, there is also something used in the airshow world called smoke oil (sometimes called "Holy Smoke Oil"). It is basically a mixture of a fine mineral oil and paraffin, so it doesn't burn, but smokes when mixed with exhaust gasses. It is usually injected into the exhaust manifold. There are a number of different systems out there, and some use less oil, but give you anemic smoke. Then there are others that give a ton of smoke, but don't last as long. I have seen some smoke systems burn through 14 gallons of smoke oil in 15 minutes!).
Rob Harrison makes some really awesome smoke systems. Here is his system on his Zlin in action: