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Like many languages, Lingála combines tense, aspect, and mood into a single TAM marking. Three of these TAMs pertain to the past:

• 
  a-kɛnd-ákí "he left earlier today" (hodiernal/recent past)


• 
  a-kɛnd-áká "he left a long time ago" (distant past)


• 
  a-kɛnd-á "he left, and he's never coming back" ("irreversible aspect")

This third tense, sometimes called the "irreversible", "ultimate", or "dead" TAM, describes an action that's happened and now can never be reversed. According to one native speaker, responding to a-kɛnd-á with "so when will he be coming back?" is a breach of conversational maxims and will get you a lot of blank stares.

I've never come across this aspect in other languages, and so I'm curious: are there other languages that have a specific grammatical marking for this meaning? And if so, is there a standard or commonly accepted name for it?

P.S. I'm ignoring vowel harmony in the examples for simplicity. What I'm actually interested in is the meaning.

I have not heard of such a thing, but I have heard of similar things in other Bantu languages. Generally, when you investigate the pragmatics and semantics of a Bantu language's tense system, you will find many subtle conditions of usage, for example "when you say it this way, you are disputing someone else's claim", or "you say it this way if you want to know where the thing went to". The things that inflectional categories encode are not at all limited to classical utterance-time etc. Neo-Reichenbachian distinctions. Sometimes the distinctions are common enough that we invent special terms (e.g. "perstitive" or "persistive", which has only recently spread outside Bantu). I have heard of this being described as "present grounded in distant past" for Lingala, and perhaps Michael Meeuwis (who works on Lingala tense) has an analysis and terminology to go with it. I just located an instance in Nurse's book Tense and Aspect in Bantu p. 113 where he calls this the "anterior" aspect, P2 tense.

A strategy that I don't entirely disagree with is to assume that the Neo-Reichenbachian approach is somewhat reasonable, and try to figure out what kind of aspect this could be, for example "perfective", and see if that predicts anything else. I would especially focus on the question of defeasibility in deciding whether a certain property is part of the literal meaning of the form.