what are the worst things about insulation construction workers?

He should be wearing PPE, that is Personal Protective Equipment that should be required by his employer.  In other words, it's an OSHA standard if I'm not mistaken that he should be wearing at the very least protection for his mouth and nose, plus for the hands and arms, depending on the insulation he's installing.  
Technically, if you perform any job using the proper PPE, you should be safe.  Other than that, use common sense.

There are two ways to answer this question based on what type of insulation construction your bf is involved in.  If he removes asbestos (asbestos abatement worker), which was used as insulation up until the 1970's, there are many health concerns.  They say it takes only one asbestos particle to be inhaled to cause asbestosis, a fatal disease.  He can also develop mesothelioma (not sure how to spell it) which is a type of lung cancer caused by asbestos.

Hopefully he does not work near asbestos.  Even if he insulates pipes that had asbestos removed from them is risky due to the fact that one asbestos particle may be deadly.

If he only installs insulation, such as building insulations / pipe insulations mostly made of fiberglass, mineral wool, calcium silicate (cal-sil), rubber, etc., there are no real health risks.  He should wear a dust mask if he is cutting fiberglass or cal-sil since the particles inhaled cannot be good for you, even though the manufacturers claim they are not carcinogens (dont cause lung cancer).  

My advice to your bf, from someone who deals with insulation every day, is to avoid asbestos at all costs.  Also, tell him to wash his work clothes seperate from his other clothing to avoid getting fiberglass splinters in his non-work clothing (can be extremely itchy).  And as he should already know, take cold showers to avoid getting fiberglass splinters in his hands/ skin.