It’s worth pointing out that uppercase and lowercase characters are mostly a quirk of the Latin, Greek, and Cyrillic alphabets. While these alphabets probably make up a plurality of written texts, many languages especially in Asia do not use these, and thus have no such uppercase/lowercase distinction.
Second, some languages may use symbols that resemble what in other languages might be uppercase/lowercase letters, but in this language actually represent different sounds or concepts altogether. Take, for example, Cyrillic В and Ь, which visually resemble Latin uppercase and lowercase B/b, but are two distinct letters with distinct and different pronunciations – and each have a distinct uppercase and lowercase form themselves. A similar example that has been pointed out in another answer is Klingon, where no uppercase/lowercase distinction is made (although the writing system is based on the Latin alphabet), and where q and Q are distinct letters.
After considering these two, the only languages to which such written ambiguities may apply are those which have rules for capitalising some words mid-sentence. As far as I am aware, this includes most if not all real-world languages that have adopted one of the scripts mentioned above. Thus, at least a vast majority of these languages should have at least one pair of words where capitalisation really matters.
German is especially rich in these, as it is a language that capitalises even common nouns – leading to cases such as der gefangene floh, which can give either der Gefangene floh (the prisoner escaped) or der gefangene Floh (the captured flea).
Many English examples have been given in various comments, of which I find helping your Uncle Jack off that horse the most amusing.
In French, it took me two seconds to come up with il est allé vers le nord/Nord; uncapitalised this is a cardinal direction, but capitalised it refers to the département du Nord including the city of Lille.
In Finnish, the town of Lahti corresponds exactly to the word for bay, lahti, except for capitalisation.
Unfortunately, I don’t speak any other languages sufficiently but I’m sure you can find examples in most as stated above.
: There are more such writing systems in minor use, of which Armenian
probably deserves an honorary mention.
: Greek to a lesser extent, but since Greek is the ancestor of both
the Latin and Cyrillic alphabets, it deserves a mention here. CJK
ideographs will easily surpass Greek and might even surpass Cyrillic
in written usage, especially given the long history and archives of
CJK ideographs. Arabic will also surpass Greek easily but might not