Plants get nutrients and water from the soil, but they don’t uptake sugar since they make their own glucose from photosynthesis. Water gets into the plant root by osmosis, and if there is too much sugar in the water, the plant won’t be able to uptake water effectively and it will wilt and possibly die. So you can’t cause a plant to get diabetes by feeding it too much sugar.

Even though plants make their own glucose they do have to carefully regulate the amount of glucose available and maintain homeostasis. In plants, sugar regulation is usually controlled by the vacuoles, which store excess glucose when the plant has plenty, and release sugar when needed. Excess sugar can also be converted to starch and stored in plastids. The signaling pathway that tells the vacuole when to release and when to store sugar is quite complicated, but it basically involves regulatory proteins called kinases, some of which bind to sugar and can sense when there is too much or too little glucose. These proteins can also signal the plant to turn on or off various genes that control the plant’s metabolism.

In humans, the regulation of sugar in the blood is controlled by the hormones insulin and glucagon. Diabetes is associated with the pancreas not making enough insulin. This is a very different mechanism of sugar regulation than that used by plants, and so plants can’t get diabetes because they have no insulin and no pancreas. But your question brings up a good point: given that sugar regulation is important and involves many interacting components, any problems, such as an important gene mutation that messes up the plant’s ability to regulate its glucose metabolism, would be very detrimental to the plant.