By "triphthongs" I assume you're referring to /eɪə, əʊə, aɪə, aʊə, ɔɪə/ found (or posited) in words like in layer, lower, fire, hour, lawyer for Standard Southern British English (or Received Pronunciation or BBC English or whatever you want to call it). And by "to master the phonology of English" I assume you're referring to acquisition of the phonology of English as a second/foreign language, not the study of it. And I'm also going to assume the accent model as a target for acquisition being talked about here is one based on what has traditionally been referred to by the names I mentioned above (SSB/RP/BBC), because if your model is based on American English triphthongs are most likely irrelevant.
If these assumptions are true, the answer I'd say is no. First of all, not all accounts of the English vowel system accept that it has triphthongs. Out of the three major pronouncing dictionaries, two (Longman Pronunciation Dictionary and Routledge Dictionary of Pronunciation for Current English) do not posit triphthongs, transcribing layer, fire, etc. as (prototypically) disyllabic and thus as rhymes with player, liar, etc. This is because it is very difficult, if not impossible, to determine whether /eɪə/ etc. in certain words are one syllable or two. Some who support the triphthong analysis point to the fact these sequences of vowels often undergo a process called smoothing—whereby the vowel in the middle is dropped, resulting in a diphthong or even a monophthong, so that fire is pronounced as [faə] or [faː] (similar or identical to far)—or argue that monomorphemic words like layer/hour/fire are pronounced more quickly and undergo smoothing more often than words like player/lower/liar, but the problem is that smoothing does occur in the latter words too and even across word boundaries. I see no reason to try to pronounce layer, hour, fire, etc. any differently from player, lower, liar, etc. If one wants to sound more like native speakers of SSB/RP/BBC, one should master smoothing (in both groups of words).