See, this is one of those things a native speaker just knows but might not be able to articulate, but I'm going to give it a go.
A type of metal button
Here, there is no metal button (well, there probably is, but the button itself isn't relevant). "Metal button" is acting as a common noun, a noun that refers to a class of things. Such nouns are generally used in the singular and don't get articles or determiners more generally. Technically, of course, button is the noun and metal an adjective, but they work together as a noun phrase filling the role of common noun.
Note that a noun being a common noun is a matter of usage, not inherent to the word itself. "The type of this metal button" is no longer a common noun usage, because there's an actual physical and specific metal button.
A case of bee sting
Again, bee sting is a common noun, a class of thing. In both of these examples, another noun is the 'main' noun of the clause - type and case, respectively. In both cases, the of-phrase refines that main noun. It is, you might say, a property of that main noun.
There was the case of a landlord vs tenant
This is a little broken, because tenant should get a determiner as well, or be plural. The sentence also works without either of them, with a subtly different meaning. "The case of a landlord vs their tenant" means you are talking about a specific case in which a landlord was facing their tenant, presumably in a legal action. "The case of landlord vs tenant" would refer to a category of cases, even though case is singular here (for complex reasons not relevant to this question - it wouldn't work for all other words that might be in place of case).
Determine a type of a lesion
This one is also broken; a given lesion will generally only have one type, so it ought to be "determine the type of a lesion". This is an instruction to determine the type of a lesion. It might be a specific lesion, or it might be telling you to find some lesion somewhere and determine its type, but in either case the action would involve an actual specific lesion.
In these later cases, where there is an article on landlord or lesion, there is, at least conceptually, a specific lesion or landlord involved. Lesion and landlord are not referring to their respective categories, but to actual concrete items.