I think there aren't any, with two tentative positive results. Of course, it is hard to prove a negative. But here are my arguments:
- Logic. there is little reason for this to happen. A culinary fruit is supposed to be sweet, and plants tend to store sugars in their botanical fruits, not in other parts (although by this logic, there should be tuber-fruits too).
- Source checking. I went through two books intended as comprehensive reference works in the kitchen, The professional chef and McGee On Food and Cooking. They don't list a (culinary) fruit which I could identify as a botanical non-fruit, with one contentious case. Note that On Food and cooking's chapter on vegetables is structured according to plant part, and it has a subchapter called "fruits used as vegetables" - but it does not have a chapter of "vegetables used as fruit" or similar.
- Language. People tend to mix up the two categories of "culinary fruit" and "botanical fruit" until the difference is pointed out to them. While they don't follow the correct botanical definition of fruit, a plant part seems to have to check the "used by the plant for reproduction" check box to be called a fruit, to the point where many words related to "fruit" evoke a harvest-like image such as "to come to fruition", or also words for "barren" in many languages which literally translate to "without fruit", e.g. the German unfruchtbar. So even when a food fulfills the "plant part" and "sweet snack" criteria, it is not called a fruit if it's not (commonly perceived as) a seed container.
The contentious case coming from the books is rhubarb. The professional chef lists it as a fruit, while McGee calls it "a tart stand-in for fruit" in the vegetable chapter, and mentions it again in the fruit chapter, as "a vegetable that often masquerades as a fruit". Kitchen terminology is by no means standardized, so if you were to question random cooks about rhubarb categorization, I guess you'd get a mixture of "a fruit" , "a vegetable" and "I don't know" answers.
An idea I had while writing the answer: maybe you can count ginger. Ginger has many uses in the kitchen, but it is available as a candied snack, which is else only common for fruit. McGee lists it as a spice, although he explicitly speaks of "the dried spice" without giving a category for the fresh rhizome. I think most cooks won't dub ginger "a fruit" but would be willing to include candied ginger in a bowl of mixed candied fruit.
Some petals and leaves are also made into jams, which would support a fruit categorization. Nevertheless, I've never heard of them being referred to as fruits. Prominent examples are rose petals, violet petals and verbena leaves (although one could argue that preserving verbena produces a herb jelly, not a jam).
This answer encompasses Western cooking traditions. It is possible that other cuisines have a different view of what a "fruit" is - for example aloe is frequently used in sweet preparations, maybe the leaf flesh is eaten as fruit somewhere.
So in the end, it comes mostly to your willingness to stretch the definition of a culinary fruit. It is certainly not the case for the most commonly eaten fruits.