In addition to any precuations mentioned or implied below, use common sense.
Although the current is usually low and the high frequency involved usually does not shock, things sometimes go a bit wrong. You may draw a spark beginning near time of a secondary voltage peak and ending at a time that the secondary instantaneous voltage is near zero, which gives some net DC charge transfer to you which will shock. This may be worse if you are grounded. The shock from this is usually minimal, but there is no guarantee of safety, especially with larger coils. Maybe nobody has ever died from this yet, but there is the chance the shock may jolt you into bumping something and breaking it, or contacting more hazardous points such as the primary circuit of a spark gap Tesla coil.
Sparks and arcs also produce nitrogen oxides which can also be hard on lung
tissue. Nitrogen oxides can be changed to other nitrogen oxides when mixed with
ozone and enough heat or ultraviolet. The worst one is nitrogen pentoxide (N2O5)
which combines with any moisture to form nitric acid. Some other nitrogen oxides
form acid solutions in contact with moisture to a lesser extent. The most
obvious nitrogen oxide is nitrogen dioxide, which is to some extent corrosive
to lung tissue and maybe rubber objects.
Nitrogen oxides may be formed in very significant quantities by the high current
sparks in the spark gap of spark gap type Tesla coils. You can largely confine
the production of noxious gases by putting the spark gap in a closed container.
This container should be either not perfectly airtight or be capable of
withstanding the pressure when the heat of the spark makes the air expand. If
you put the primary spark in a glass jar, you may see the air turn orangish
brown from the nitrogen dioxide. Production of nitrogen dioxide will stop when
nearly all the oxygen in the jar has combined with nitrogen.
One really oddball hazard is any effects of putting your head in a strong enough radio frequency magnetic field. This is more a hazard of solid state coils which oscillate continuously, and probably requires being at a close range (maybe almost having your head inside the secondary). The problem is effects of a sudden temperature change in your brain. This may cause a seizure; in milder cases suddenly reduced mental function. In severe cases related to point-blank exposure to powerful microwave beams from radar transmitters, people have been known to pass out and stop breathing - obviously dangerous! Severe effects can occur from a sudden temperature change of only a degree or two of brain temperature.
One more thing - although having someone nearby who knows CPR or who has a defibrillator handy is good, the success rate is less than 100 percent.
Conductive objects in the coil's electric field can make sparks if they touch or get very close to other conductive objects that are grounded or unequally exposed to the coil's electric field. These sparks could start fires. One fear I have is that a stray wire, pipe, pipe support strap, or a long nail behind a wall could touch some damp wood behind the wall and make a spark and then a carbonized spot and then start a fire.
Closed loops of fine wire may burn up if they are close enough to the coil to get enough magnetic field to have a lot of current flowing through.
The strong AC electric field is probably not too good for electronic equipment, especially if the equipment is running. This is particularly true with computers! They may "latch up" and have even been reported to spontaneously reboot when exposed to the electric field of a Tesla coil.
NEWS 7/6/2005 - I have caused an answering machine to falsely detect an actually nonexistent incoming call when 5 feet (1.6 meters) from a slightly large size Tesla coil that developed peak voltage of merely 40, maybe 50 kilovolts!
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