For context, it's worth stating that no system for writing language has long remained purely logographic (though a set of logograms could constitute proto-writing).
A complete writing system for a language needs to have methods to indicate not only concrete concepts that you can draw a picture of, but also grammatical and abstract words, many of which are difficult to represent logographically. What we see early writing systems do is repurpose some originally logographic symbols according to the rebus principle: the logograph is used for the sound of its associated word, and thereby represents another word that is pronounced the same or similarly (like if we used 👁️ to represent the English first-person pronoun).
The use of symbols for their phonetic value is seen in Sumerian cuneiform, Chinese characters, and Egyptian hieroglyphs. Thus, all of these are mixed logophonographic writing systems, not simply logographic writing systems.
As Draconis brought up in the comments, few languages can be said to have entirely phonological writing systems either. But of course, some writing systems have drastically reduced logographic components compared to Sumerian/Egyptian/Chinese. Switches from a system with a logographic component to a nearly completely phonological writing system seem to generally be facilitated by the process of borrowing a script to write another language that it wasn't originally used for, so I'm not sure whether the "smooth transition" within one language brought up in the comments is a well-attested development if you're envisioning the endpoint as being a writing system like modern English, Greek, Arabic, etc.
Andréas Stauder (reviewing The Shape of Script: How and Why Writing Systems Change, by Stephen Houston, 2012) writes
The hieratic script and the demotic one (which historically derived from the former by around 650 BCE) represent the same language, Egyptian. On an extremely general typological level, both scripts are logo-phonographic, and shifts at this level may indeed be limited to contexts of adaptation to another language (a further instructive case being the rise of the Meroitic script, an abugida, out of the phono-logographic Egyptian scripts).
(Another source of phonological writing systems is their de novo creation using symbols with values not directly adapted from a prior writing system; this process seems to have occurred historically only in contexts where the creator was already exposed to the general concept of writing: e.g. the invention of the Cherokee syllabary by Sequoyah.)
However, there are certainly examples of mixed logophonographic writing systems evolving to show increased use of signs for their phonological value without fully abandoning logograms.
In the case of Egyptian, Wikipedia briefly suggests that the latest stage of its writing system showed a greater proportion of phonological spellings:
Like its hieroglyphic predecessor script, Demotic possessed a set of "uniliteral" or "alphabetical" signs that could be used to represent individual phonemes. These are the most common signs in Demotic, making up between one third and one half of all signs in any given text; foreign words are also almost exclusively written with these signs. Later (Roman Period) texts used these signs even more frequently.
(Demotic (Egyptian))
The Coptic language is a descendant of Egyptian, but the Coptic writing system is an adaptatation of the Greek alphabet (augmented by a handful of additional letters). While the Greek alphabet is itself largely derived in the end from Egyptian hieroglyphs, this is not an example of "a primarily logographic writing system becoming entirely non-logographic over time" unless you meant to include the detour to Greek as part of that process.
