Sign languages generally do not have rich case systems because they tend to be much more head-marking than, say, English. By this I mean that a translation of your Latin sentences into a hypothetical sign language might be something like
BOY FARMER KILL-he-him
him are verbal inflections that make it clear who killed whom. If we consider word order, then ASL and BSL both are like Latin: they have default word orders, but a sentence might deviate from it (for example to highlight the boy or farmer, or a contrast from an earlier sentence), which means that word order itself cannot unambiguously distinguish, for example, agents from patients, like English word order usually can. But that doesn't matter so much, because the distinction is marked on the verb. For this reason, verbs don't really need to assign case in signed languages.
As a rule, if verbs act this way in a given language, then noun phrases tend to be more head-marking too. It's rather like how Hebrew, Berber or Ainu do possession and this appears to be a linguistic universal. So for instance, "the farmer's boy" might be translated as "the farmer, his boy" in both ASL and BSL. So here there doesn't need to exist a genitive case either*.
That said, a pronouns are a slightly different consideration. For example English and Norwegian are both languages where pronouns have two or more forms, which have something to do with case, even though the nouns do not productively inflect for case. ASL is the same! Here are some examples from ASL:
1 handshape (extend the index finger but clench other fingers together into a fist) and point towards any person is for a subject or object pronoun, like I, me, he, they, them, etc.
B handshape (all four fingers straight out and parallel and touching) and point towards any person is for a possessive, like my, mine, his, their, etc.
B handshape, accompanied by a downward movement is an alternative to
1, which is rather like a T/V distinction except it may be used in any person, including 1st.
And I'm not aware of any other inflections in this morphological class.
*: Some constructed sign languages are invented for didactic reasons and they artificially include morphology and most word order from the culture's spoken language. An example of this is SEE, which means Signed Exact English. These tend to be falling into disfavour now. I didn't describe these in my answer because I feel they are better described as a manually coded version of a spoken language.