Some languages change what writing system they use. For example, Old English used to use Anglo-Saxon runes but eventually used the Latin alphabet, and Mongolian in Mongolia uses the Mongolian Cyrillic alphabet (while in China they still use the Traditional Mongolian alphabet). Likewise, some languages used to not have any writing at all.
Does the change in writing system change the language, such as how words are pronounced?
There's a long list of languages which changed their orthography (maybe even several times). Sometimes gradually and sometimes through reform. Some of those changes were relatively small like the move from Schwabacher in German and some other languages, or introduction of diacritics as in Czech. Sometimes quite major such as the change from Arabic to Latin based script as in Turkish or from Cyrillic to Latin alphabet in Turkemnistan (having previously switched to Cyrillic from Arabic script), or Korean switching from Chinese characters to its own orthography, etc. You could also think about the introduction of intermediate scripts in the teaching of Chinese writing (such as Pinyin or Bopomofo) or think about the fact that most speakers of languages with orthographies other than Latin are often by-scriptal - just through exposure to things like advertising or the Internet. You also have situations where the same language is written in different orthography depending on location (such as Turkmen being written in Arabic script outside of Turkmenistan). You also have cases where the only (or the main) distinguishing mark between official languages is the official orthography (Serbian and Croatian or Moldovan and Romanian).
So we have ample evidence that the introduction of a new system of orthography does not change the language (in as much as that is even a sensible way to formulate it). But it does have an impact on the community of speakers of the language in question (and as jknappen points out, is often accompanied by other changes). One of the avenues of impact could be simply that the speakers of the language all of a sudden have access to different writing traditions and that subsequent generations will have more limited access to the previous writing tradition of the language.
On the other hand, it has been hypothesized that the introduction of writing to a language does have an impact on the development of syntactic features such as sentence and the formalization of things like genre. However, that is not due to orthography but rather the very nature of writing which is not just transcribing what is said but creates its own ways of saying things. So introducing writing to a language is very different to changing an existing orthography and is likely to have some impact on the language - if only through borrowing of structures of other languages - as in the translation of the Bible which has been one of the first written text in many previously un-written languages.
There is a plain answer: No, it doesn't.
Historically, only a minority of speakers were literate; therefore the spoken language did not follow the writing. Even in societies were almost all adults are literate, the influence of the writing to the spoken language is rather small.
But: A change of the writing system often comes with massive other changes in the speaker's community (like Atatürk's reforms, a foreign invasion, declaration of independence, changing of military and cultural affiliations, success of a new religion) that have massive impact on the language on all levels.
So a change of the writing system often coincides with a change of the language, but it does not cause it.
If it’s not the case I’m glitching…
There`s some phenomenon which can be named ‘orthographic(al) pronunciation’, a kind of hypercorrection based on received spelling. (The term is coined in Russian (‘орфографическое произношение’), AFAICS, to denote the impact of spelling on pronunciation.) But all this is the case of an existing, not new but maybe spreading through education, writing system. With an impact – I’ll be surprised if worthy of the 'language change' status.
I’d be even more surprised if such phenomena are observed in relation to changing writing systems.
It depends, in most cases it doesn't change a language as mentioned in other answers. However, here are some cases I believe it makes a (very mild) change to the language, especially in its lexicon of the written language:
Examples of spelling affects pronunciation as mentioned in another answer: GIF, often (silent t or not)
Chinese, Japanese and Korean As people answering this question do not use an logographic writing system. Here are some important additional points:
I don't speak Korean but Japanese and Chinese. Similar to Japanese, there are many homophonic Sino-Korean words (mostly formal words) mainly used in written text but not in spoken language. Those words are now written in Hangul without Hanja for disambiguation most of the time (Context disambiguates most of the time). I wonder some homophonic Sino-Korean words are used much less now due to the confusing nature of the homophones, or it still needs Hanja for disambiguation.
If Japanese/Chinese are written without Kanji/Hanzi, it is entirely possible to read. But some old expressions/ technical terms in the written language along with some names are homophonic in nature and they will be lost (or they still need Hanzi for disambiguation).
eg. 日本製紙グループ株式会社 (製紙 = paper manufacture and it sounds like another word 精子 = sperm)
Some other terms may be used instead to avoid the awkward term.
Similar case in Chinese. Old Chinese poems and classical Chinese quotations and idioms and etc are used much more frequently in written forms (and in daily conversations) than many other languages (eg. English). Those idioms are sometimes quite homophonic (think about the popular poem with the syllable "Shi") and once the Chinese characters are abolished, those expressions will definitely be used much less to avoid confusion (it's very hard to read).
In addition, Chinese is the most loanword-resistant language in the modern world because it uses Hanja. As the Chinese consider only words written in Hanzi "standard Chinese" with very rare exceptions like eg. USB. Even Wifi and coronavirus has a calque written in Hanzi. Long phonetic transliteration is awkward for Chinese as it results in long meaningless Hanzi combinations which is very hard to remember (Hanzi is very information dense. they are mostly morphemes representing meanings. You don't use a logographic writing system to transcribe sounds!). It is the writing system that makes Chinese having so few loanwords (many "code-switching" in spoken Chinese but they are not considered "Standard written Chinese" as those sounds are not part of the written Hanzi and cannot be written with Hanzi).
Changing Hanzi to an alphabet (eg. Pinyin) would make Chinese more accepting to loanwords and introduce new sounds to the language (as more and more Chinese speak English as well) and it may change the future lexicon completely.
