Disclaimer: I am not a linguist, please provide any corrections for terminology.
From How languages compare with the number of different syllables from all words?, Yoon Mi Oh's thesis counted the different syllables in the 20,000 most frequent words for several languages. Her results, ordered by increasing number of syllables are:
Japanese: 643 Korean: 1104 Mandarin: 1274 Cantonese: 1298 Basque: 2082 Thai: 2438 Italian: 2729 Spanish: 2778 French: 2949 Turkish: 3260 Catalan: 3600 Serbian: 3831 Finnish: 3844 Hungarian: 4325 German: 5100 Vietnamese: 5156 English: 6949
We can see all languages with a writing system historically based on (approximately) syllabaries (with characters associated with syllables) — Japanese, Korean, Mandarin, Cantonese— have considerably less different number of syllables than others based on alphabets. Note that Korean until the end of the 19th century used Hanja, a writing system mainly based on Chinese characters. There is also an exception: Vietnamese, with a large number of different syllables and using until the 20th century Chữ Nôm, also a written system based mainly on Chinese characters. Despite this case, can a language restrict overtime its number of distinct syllables because it is written with a syllabary?
I only have an idea for the case of Mandarin. Although there are many Chinese characters, literate individuals know and use between 3,000 and 4,000 characters (see 3). Maybe this is the physiological upper limit on the number of items of a syllabary for an average person to remember? In that case, this would place an upper bound for the number of different syllables for a language using a syllabary. Furthermore, Chinese characters are constructed by combining only 214 radicals, which sometimes are responsible for its phonetics. And the 10 most used radicals appear in 10,665 characters (or 23% of the dictionary), potentially restricting the phonetics even more.
Supporting the direction of the hypothesis (from San Duanmu, The Phonology of Standard Chinese):
While Middle Chinese (about AD 600) had over 3,000 syllables (including tonal distinctions), modern Standard Chinese (SC) has just over 1,300. Thus, over a period of 1,500 years, Chinese lost more than half of its syllables. Moreover, the syllable inventory of modern Chinese continues to shrink. In addition, about 200 of the 1,300 syllables in SC are now rarely used.
In addition, Evelyn Rawski writes that during the Qīng Dynasty (1644-1911):
Evidence of the large number of potential teachers and the widespread distribution of private schools led us to conclude that it was possible for a broad cross-section of Ch'ing [= Qīng] males to attain some degree of literacy in private and charitable schools. Information from the mid and late nineteenth century suggests that 30 to 45 percent of the men and from 2 to 10 percent of the women in China knew how to read and write. This group included the fully literate members of the elite and, on the opposite pole, those knowing only a few hundred characters. Thus loosely defined, there was an average of almost one literate person per family.
Thus, a large part of the population was literate, which could allow for some plausible influence from the writing system to the language.
For the particular discussion of Chinese language, I also asked the question: Why Mandarin Chinese has a few number of different syllables?