Does changing the writing system for a language change the language?

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.