Can plants grow without photosynthesis?
To answer if the equivalent could be done with plants, we have to understand the histological and anatomical differences between plants and animals.

Just as spraying a nutritive solution over an animal (like you and me) doesn't work, spraying a nutritive solution over the leaves doesn't work as well: the nutritive molecules (let's use glucose in our hypothetical example) don't penetrate the leaf's epidermis. The same happens if you put the nutritive solution in the soil: there are several membrane layers between the external medium and the phloem, and glucose will never reach its destination.

Now, let's try to compare with a IV injection of nutrients, which we can do in an animal. The plant's structure that (kind of) can be compared to blood vessels is the phloem (actually, xylem and phloem, but in our case, dealing with nutrients, we're gonna talk only about the phloem). To inject the nutrients we have to reach the phloem, because the periderm in plants with secondary growth ("cork" in the image) is very impermeable:



Now, even if we reach the phloem, using methods like this...



...we still have a problem: in an animal like you and me, a substance injected in a given vein will quickly spread to all the systemic blood vessels. But the same doesn't happen with the phloem. The injected substance can go up or down (depending on the pressure flow), but it doesn't spread to all of the plant's tissues, unless we perforate the phloem in a lot of places.

Thus, so far, this method is used to deliver chemicals (like pesticides) into the tree. Notice that, just like a person in a hospital, the bag with the solution has to he higher than the perforation:



For a chemical treatment like this, the amounts of the substance are small. But for "feeding" a plant using this technique, the amounts of necessary nutrients are way higher...

Finally, answering your question: is it hypothetically possible to feed a plant in the dark with a nutritive solution? yes. Is it practical, or doable? very hardly.

Source: Utah State University