Once an onomatopoeia, always an onomatopoeia (usually).
Although the meaning of a particular onomatopoeia may develop from a mere verbal imitation of a sound to a fully fledged word with multiple detailed denotations, it remains an onomatopoeia so long as some semblance to the original imitated sound remains. According to Merriam-Webster, onomatopoeia is "the naming of a thing or action by vocal imitation of an associated sound". By this definition, an onomatopoeia need not merely represent the sound itself; verbs and other parts of speech can be onomatopoeic, provided they derive from a related sound (in nuce, words describing things or actions can still be onomatopoeic).
One typical instance of an evolving onomatopoeia is the history of the word hiccup, which incidentally is given as an example of onomatopoeia in the Merriam-Webster entry. Over time, hiccup evolved from a pure imitation of a sound (Hiccup!), eventually referring to an instance of making that sound (he made a hiccup) or the process of making that sound (he hiccuped), but it still maintained its essential imitative character.
Although speakers may eventually forget the imitative origins of words (especially as their meanings evolve), some words still accurately imitate their associated sounds, even after switching languages. Take cliché, for instance: cliché comes from the French word for a literal printer's stereotype, which in turn derives from the verb clicher, an imitation of machinery forging a stereotype (see Merriam-Webster and Wikipedia for etymology).
Even piss (which, according to Merriam-Webster, ultimately derives from the imitative Vulgar Latin verb *pissiare) is still essentially onomatopoeic. Although various vulgarisms with myriad meanings derive from piss, they all (at least historically) relate to urine; crucially, the word piss still sounds remarkably like the timeless act of urinating, all these centuries after the word's ancient Roman origin. (Similar arguments could be said of "choo-choo", "ring", and other historical onomatopoeia.)