Do we know of any cases where the grammar of a language was influenced by the imperfection of its writing system? For example, has any language become isolating because it had a logographic writing, which didn't express inflections?
This is really not the right question to ask. Writing is not just about the way of writing things down but also its own way of communication. People who read and write, don't just transfer speech into letters. They express themselves differently in the two modes. This is partly because of the different cognitive demands of reading/listening and partly because the two are used in different contexts. But in effect, all literate people are to an extent bilingual - they are competent in two codes and they switch between those codes with ease.
But as in all instances of language contact, there is cross-pollination. So it's quite likely that complex multi-clausal sentences with embedding and hypotaxis developed in complexity alongside writing - although I don't think there is any definitive proof of that. In some cases, written language perpetuates older forms - e.g. in Czech or Arabic that would probably otherwise disappear. In English, there are many pronunciations influenced by spelling and vice versa. None of these examples constitute 'imperfections', just features. Although, it would not be difficult to argue that the English spelling system is completely flawed, the impact of these flaws on spoken English has been minimal.
I don't know of any case of a particular script or orthography influencing grammar but writing in general can have an influence. Written forms of a language are very often their own "dialect" or register. Sentences tend to be longer and more grammatically complex. Undoubtedly this feeds back into the spoken language to some degree. But I guess this isn't the kind of answer you are looking for. I think it is more likely that grammar will influence the writing system.
No, Chinese characters or what we call Chinese logogram have not influenced it's grammar at all. Writing is just a image of the mapping from sound and maybe meaning of language. How could it influence the rule of language? From modern model of evolution of language like evolution game theory, also it is unlikely that logogram influences grammar either!