I've heard that Tasmanian Blackwood has toxic sawdust. Just how toxic is this sawdust, and what precautions need to be taken when working with it?
Sawdust in general is very bad for your health. The smaller bits that float around (especially from sanding) become little particles that act just like asbestos, they can get into the alveoli and damage them. So right there is a good reason to have a dust mask and/or a good dust collection system.
There are many woods that have toxicity on top of the physical danger of sawdust. Generally the darker or more aromatic the wood the more chemicals that can react with your body. Walnut is full of natural herbicides and pesticides and they don't do the body good. Some are just allergic reactions so some people will be affected and others won't.
Walnut and your Blackwood both have a 2 star rating according to the wood database link saltface put in the comments. Having used walnut I would be very hesitant to use a 3 or 4 star wood without very good precautions!
It's impossible to say, really. You may be allergic to a wood that another person isn't. For most woods they're little more than mild irritants. Walnut doesn't bother me for example, but Wenge makes me break out in a horrible rash akin to poison ivy that lasts for weeks. At a minimum you should wear a dust mask for ANY wood . I got so bad with wenge that I wear a full face mask respirator from 3m and a tyvek hooded suit with neoprene gloves. You can't really know until you're exposed, and adjust your protective equipment accordingly if you have a reaction. Be aware that some woods may not cause a reaction on the first exposure but will get progressively worse the more you work with them.