Has an FPGA ever caused health issues?
I'm no expert, but I'd postulate the following:


Your biggest risk is inhalation of fumes from soldering. The materials involved (e.g. lead) are often hazardous, and flux fumes are known to cause irritation of the lungs.
If you're directly involved in chip fabrication, silicon dust is another hazard to avoid, since it can lead to silicosis if inhaled.
Many component casings are made of polymers, which might give off some nasty gases if burned. You're probably looking at molecules containing halogens (e.g. chlorine), which may be hazardous if breathed. However, you're unlikely to release more than trace amounts of these unless you actually burn the case directly with the iron.
Long-term exposure to smoke and other chemicals on your fingers may cause hardening and discolouration, but this is highly unlikely to be damaging to your health in general.
An appropriate fume extractor should be enough to protect you from most problems.


All in all I don't think there are any risks that are specific to ASIC / FPGA handling.