There are many kinds of incense, some made with all natural, partially natural or completely synthetic ingredients. And while there have been many speculations and opinions as to the potential negative impacts of burning incense, there haven't been any scientific, peer reviewed studies to determine the effects that smoke from burning incense has on our health.
Researchers who work for a tobacco company conducted a small study which, as you can imagine, was very biased and seemed to be done in an attempt to shift the negative attention away from cigarettes, to something that there is no evidence for.
Civilizations all over the world have been burning incense for thousands of years, and we have yet to see a case, anywhere in the world, where incense has been blamed for health problems or deaths of a bunch of people, if any at all. The way we burn incense today hasn't changed that much from the ancient times.
In saying that however, I believe it is important to always use common sense and when you do burn incense (including smudge sticks or loose leaves, herbs and flowers) that you should make sure the space is well ventilated, especially if you have a lung condition such as asthma for example.
Some people have claimed to have been affected negatively by incense, oil burners and candles that use synthetic ingredients however that is purely anecdotal. That is not to say that they did not have an effect, they very well may have, however we are not sure if it was due to the incense alone, or that it's an effect that happens all the time.
So, until there is a sudden outbreak of health problems or deaths that all have incense in common, it is safe to assume that, at least in my opinion, burning incense in well ventilated areas is pretty safe. Just take extra precautions if you have lung issues.
Love, light and blessings.