Is working MDF bad for your health?
Yes but only if you are ignorant of your tools and materials.
It certainly can be. However the answer of "Yes" is misleading as the concerns surrounding MDF are more than just the potential presence of hazards such as the release formaldehyde. Working MDF creates dust particles much like any other wood. It is true though that some formaldehyde will be located on those particles and that has the potential to enter the body.
Articles and blogs that discuss the harms of formaldehyde usually miss some key points. Namely that it is a naturally occurring organic compound that is found in the air, plants, animals and our own bodies.
Formaldehyde is normally present at low levels, usually less than 0.03 ppm,(parts per million) in both outdoor and indoor air.
From www.hse.gov.uk: What is formaldehyde and where does it come from?
Large point to make about MDF is that the common binding resin urea-formaldehyde, for which the concern derives from, is not the sole binding agent used in the manufacture of MDF today. Many suppliers offer formaldehyde free or no added-formaldehyde options.
Another thing that some people forget is that the compound urea-formaldehyde has also been used in the creation of plywoods and other sheet goods. Therefore its use in the engineered wood market is not new.
Should I avoid its use?
There are certain safety precautions that need to be taken regardless of the work you are doing. I like this quote from a prop makers blog that captures this well:
Like any other substance or material used in [the workshop], the safety of using it is dependent on knowing the risks and possible hazards and taking the appropriate precautions.
In that video that was linked I don't think I see a mask at all in use. Sawdust is a known carcinogen from MDF or otherwise. As discussed in What kind of wood dust is toxic/dangerous and requires usage of dust mask? the answer is you should always be wearing the appropriate dust mask.
Also need to be aware that reaction to different materials can be individual. Some people are just naturally sensitive to dust and particles. That does not mean MDF is a bad material on that justification alone.
Is working with MDF going to kill me?
Probably not. As long as you are aware of the risk associated with working on it. That last sentence should ring a general warning that you should always be aware of the risk to your personal safety when working with your tools and materials.