In German, some verbs use the dative case for the object (e.g. Ich folge dir). In lists about these, I've seen mention of jemand and etwas. Does the case change depending on whether the object is a person or thing?
No, the case does not depend on whether the object is a person or a thing.
One can say:
Ich folge dem Mann. [jemandem, Dativ]
Ich folge dem Auto. [etwas, Dativ]
This is similar to English dictionary entries, for example:
to follow so./sth.
where the distinction between someone and something is unrelated to cases as well.
The distinction between a person and a thing is made, because there are some verbs that usually do not make sense with a person or a thing. This is rather about word usage, less about grammar.
Consider the German verb weh tun, for example:
Dem Mann tut der Bauch weh. [jemandem, Dativ]
Dem Auto tut der Bauch weh. [etwas, Dativ]
The second example is perfectly fine from the grammatical point of view. However, it does not make sense in a standard environment, since cars usually do not have a belly that might hurt them. (In a non-standard environment, like a fictional writing or TV show, a sentence like the second might be valid as well. For example, in a story where toys are "alive", the doll might tell the teddy bear Dem Auto tut der Bauch weh., referring to the toy car.)
In German, it is a trait of a verb what case (nominative, accusative, genitive, dative, all are possible) it rules. This does not change depending on the object's gender:
Ich folge dem Mann
Ich folge der Frau
Ich folge dem Kind