I don't know of an example where every possible Chinese syllable is listed. I doubt such a text exists, because even without counting tones, there are over 400 hundred distinct syllables in Standard Mandarin. If you include tones, there will be more. Some syllable-tone combinations aren't possible, so the entire inventory is going to be massive and a little disorganized, including marginal syllables such as 誒 (éi) and 得 (děi), which makes it very hard to permute into a meaningful text.
Moreover, I interpret "pangram" a little differently from "passage containing every possible syllable", because it seems to be more about listing the symbols used in a writing system. English pangrams like "The quick brown fox..." goes through the alphabet without including all possible sounds in English. (/ŋ/ is missing for example.) One could argue the same for Iroha, which doesn't include the voiced stops that probably existed by the time it was composed.
All this is to say that I am going to be a little liberal with the definition of "pangram". So the following are some examples of pangram-adjacent compositions.
This is a poem composed in the 6th containing 1000 distinct Chinese characters, arranged into 250 rhyming 4-character lines. Intended for children learning to read and write, it works both as a literacy tutorial and a primer on cosmology, history, society, and ethics.
Since there are way more than 1000 Chinese characters, this certainly isn't an exhaustive list. It is also intended to teach the writing system more than pronunciation, since it contains homophones, like 皇 and 黃, and is missing many syllables (there are about 2000 distinct syllables not counting tones in Middle Chinese at the time of composition). In spite of of all these, this might be the closest in spirit to Iroha we will get in this list.
早梅詩 and other 聲母詩
Traditionally Chinese syllables are analyzed into 聲母 (onsets), and 韻母 (rhymes). Depending on the period, variety, and how you're counting, there are usually about twenty or thirty something onsets, and thirty something up to about two hundred rhymes. So naturally, it's easier to create a pan-onset-gram than a pan-rhyme-gram.
早梅 (the title translates to "Early Plum Blossoms") is an example of pan-onset-gram from 15th century. It was written by 蘭茂 for his rhyme-dictionary 韻略易通 as chapter headings. It goes like this (I've put the onset each character represents in parentheses),
東 (t) 風 (f) 破 (pʰ) 早 (ts) 梅 (m)
向 (x) 暖 (n) 一 (∅) 枝 (tʂ) 開 (kʰ)
冰 (p) 雪 (s) 無 (v) 人 (ʐ) 見 (k)
春 (tʂʰ) 從 (tsʰ) 天 (tʰ) 上 (ʂ) 來 (l)
This of course reflect features for the speech of the time and is different from modern Mandarin. For example in Standard Mandarin 向 and 雪 has the same onset, and palatalized /ʨ/ /ʨʰ/ /ɕ/ are not included in 早梅詩.
What's also notable about this poem is that it conforms to the meter of classical Chinese poetry, having the correct tones and rhyming, much like Iroha which follows the Imayō form of Japanese poetry.
Such pan-onset-grams have been parodied many times by later linguists. 李汝珍, who also wrote novel 鏡花緣, had in his linguistic monograph 李氏音鑑 the following poem,
Modern examples that reflect modern Mandarin phonology include 王力's
by 周有光 (one of the inventors of pinyin).
The latter also has a pan-rhyme-gram, which as I noted is harder to compose, so remains the only example I can find for now.
天子聖哲 and pan-tone-grams
天子聖哲 was a famous quip recorded in the canonical history of Liang dynasty (502-557), said by 周捨 upon the emperor's query as to what the four tones are. At that time, knowledge of tones was relatively new, so when a treatise on the topic came out, the emperor was justifiably curious and confused. The response not only demonstrates the four tones, where 天 is 平聲, 子 is 上聲, 聖 is 去聲, and 哲 is 入聲, but also flatters the emperor by praising him with "the Son of Heaven is sage and wise".
The number of tones in Chinese is much fewer than onsets and rhymes, so it's must easier to come up with a phrase that contains all. Canonically Chinese is said to have four tones, 平上去入 (which itself is a pan-tone-gram), but this probably only reflected the reality in the mid-first millennium. (Of course Standard Mandarin has four tones, but those don't match the traditional four exactly.) However, since Chinese seems particularly amenable to four-syllable phrases, this turns out to be a perfect linguistic coincidence. Such phrases are so numerous that someone did a search for them, counting only the ones that have the four tones in the standard order, and came up with more than one hundred of them. The same is true for the equivalent in Standard Mandarin, where some fun/silly examples are 千奇百怪, 雞鳴狗盜, 諸如此類. Keeping in mind your intention of using pangrams to practice pronunciation, perhaps saying these phrases could be good exercise for people learning Chinese tones.
A couple of marginal examples
One example that more or less fits the "enumerating components of the writing system" definition is the character 永, which is said to contain all the strokes that make up Chinese characters. This has been turned into a methodology for Chinese calligraphy, which dictates the proper shape and aesthetic of each stroke, and by extension the whole art of writing.
I'm also aware of a pan-onset-gram made up by linguistics hobbyists in recent years that gathers together onsets of the Middle Chinese of 切韻. It goes like this,
These representative characters ultimately came from the thirty or thirty-six characters used by linguists from Tang and Song dynasty. They are adapted according to modern analysis of the phonology of the language recorded in 切韻, so it splits the traditional 照穿床審禪 into 莊初崇生俟 and 章昌船書禪, and the 喻 onset into 雲 (whose homophone 云 is used instead in the passage) and 以. Given that this is a very specialist creation, perhaps it's becoming more of a linguistic game than a useful mnemonic.
A text that's often mentioned beside the thousand character classic is 百家姓, or Hundred Family Surnames. It's a list of some five hundred common Chinese surnames, also arranged into rhyming 4-character lines. Once again, the text is mainly an instructional book for children. While not a pangram in either the phonetic or glyphic sense, it does serve as a collection of domain-specific words.