Is it illegal to use "pharmacy" or "laboratory" words in the name of an LLC or website that is not related to a real pharmacy or laboratory? i.e. using "healthlaboratory.com" and "health laboratory" brand name for a supplement company.
Usually the corporation registration law will not prohibit the use of either of these words (there are some words related to banking and the appearance that the entity is a government that often are regulated).
The word laboratory is rarely regulated and can encompass a multitude of senses of that word.
The word pharmacy would often be restricted, not in the corporate registration law but in the pharmacy regulation law, unless it is used in manner that clearly indicates that it isn't a true pharmacy, e.g. "Fashion Pharmacy" or "Marriage Pharmacy."
Needless to say, you still can't use a trademarked name that another company already has like "Walgreens Pharmacy" or "Underwriter's Laboratory" in a place to which the trademark extends.
If you create a possibility of genuine consumer confusion, you could face liability if you mislead someone into thinking it is a particular kind of laboratory or pharmacy under deceptive trade practices laws.
Of course, different jurisdictions have different rules and your mileage may vary.
Yes, but ...
There are lists of words and expressions that require permission to be incorporated into a business or company name. Neither "laboratory" nor "pharmacy" are on those lists.
However, it is unlawful under the Australian Consumer Law to engage in deceptive or misleading conduct in trade or business. Using "pharmacy" or "laboratory" in your name might do this - it will depend on the context.
For example, calling your cocktail bar "The Mix Lab" is probably fine; calling yourself "Sydney Pharmacy" when you aren't a pharmacist probably isn't.