That's just plain incorrect about the propensity to react; BE is fairly inert. Yeah, oxidation will occur, like with most metals, but smoke? No.
Hopefully many of the comments here can be classified as overreactions, as they would suggest even looking at beryllim wrong might develop into cancer. I've handled a number of pure beryllium parts in the past few years, including aggressive cleaning with wipes and mild solvents. No hazmat suit, but I was in a cleanroom and wearing a mask and gloves. As stated, the risk is with inhalation of dust or fumes, so yes, proper precautions need to be taken in a machining environment. Vacuums with HEPA filters, along with PPE should offer adequate protection. Any particles created from machining operations will be heavy and tend to settle quickly. The smaller the particle, the higher the risk, so I would feel less concerned about cutting chips than, say, sanding, grinding or welding. Yes, fumes are a concern if the metal gets hot enough. The consequences of inhalation can indeed be very severe, but if handled with some care and common sense, the probability of inhalation can be pretty much eliminated.