As fas as I know, there isn't a single plant on Earth that (naturally) produces any sort of luminescence. But it is seen in science-fiction worlds, especially Avatar.
From an evolutionary point of view, what would be the benefits for a plant from being bioluminescent?
It is easy to think of evolution as only being about benefits, but in order for something beneficial to be naturally selected it must first arise spontaneously. The fact that plants have not so far evolved bioluminescence does not indicate a lack of benefit. It simply means that they have not stumbled upon a way of doing so.
Some bioluminescent organisms do not have any obvious benefit from their light. For some it is apparently a side effect rather than a useful trait - of interest to us but of no consequence to them. For example bioluminescent fungi are sometimes hypothesised to use the light to attract grazers to the fruiting bodies, other times hypothesised to use the light to repel grazers from the mycelium, these apparently contradictory hypotheses seeming somewhat inconclusive. However, the chemical used to produce the luminescence works by being oxidised, which means it acts as an antioxidant. The linked page on bioluminescent fungi mentions research suggesting that the production of the chemical is linked to the metabolism, and that it may provide:
antioxidant protection against the potentially damaging effects of reactive oxygen species produced during wood decay
So the fact that humans are sensitive to the particular light emitted may not be relevant to the benefit it provides as an antioxidant. As octern points out in the comments on this answer, the term for such byproducts independent of natural selection is spandrel.
If you want to populate a world with glowing plants and ornate patterns of light, they may not necessarily have any benefit from it - it would be perfectly realistic for such plants to arise purely accidentally. The light can arise and persist without need for benefit. Patterns can also arise without need for benefit, as can be seen from the various patterns of leaf veins (here the pattern is beneficial in that it is efficient, but its aesthetic appeal is incidental, not due to any visual benefit).
A plant may happen to develop a useful chemical process that has a luminescent chemical as an unused byproduct. This may be more concentrated in the veins, showing up the vein structure in glowing detail through the translucent leaf membrane. Alternatively the luminescent chemical may be a byproduct of producing a hormone. If the hormone stimulates growth then the parts of the plant currently growing fastest will glow brightest, showing the patterns of growth like a textbook picture. If the hormone promotes flower production then it may only be new buds that shine brightly, with a rapidly diminishing glow along the stems leading to them.
Some plants produce chemicals in response to damage. If this becomes linked to a luminescent byproduct the result could be leaves which glow faintly when bent or torn, and shine more brightly when crushed or shredded.
You can find plenty of examples of benefits of glowing by looking at the examples we know of, and there are doubtless many other possible benefits that could arise that we don't have examples of. However, I wanted to emphasise just how much is possible even without any benefit at all. The rich diversity of life is due to this tendency to branch out and find new forms, even if they are not immediately useful. Sometimes a beneficial trait is only possible because of a series of neutral changes that came before. Sometimes a trait we happen to find useful or beautiful has no benefit at all to the organism we find it in. I find this fact in itself to be beautiful.
Going in the opposite direction, a glowing forest scene in a fictional world may be the result of bioluminescent parasites and infections, treated as magical by the inhabitants but actually a sign of poor health of the forest, and not in any way beneficial.
The Wikipedia page for bioluminescence lists a number of known benefits for existing organisms. None of these organisms are plants but I will consider some of them for their potential as a hypothetical plant benefit.
Some deep sea fish glow underneath so that predators below them cannot sea them against the uniform glow of the light filtering down from the sky. I've been imagining land based plants but there are of course ocean plants. However, they are not present in the deep ocean where the light levels are low enough to make this approach practical, so this benefit is unlikely to apply to plants.
Again, the real world examples are ocean based - predators mimicking much smaller creatures to attract prey, but a land based plant could use bioluminescence to mimic for reasons other than hunting. There are plants that have flowers resembling insects, so that insects trying to mate with the flower inadvertently spread pollen. There are also insects that use bioluminescence to attract a mate. So it would be feasible for a plant that could glow to mimic a glowing insect in order aid pollination.
In addition to mimicking insects to attract pollinators, glowing could also be beneficial simply to attract pollinators without trickery. Patterns of luminescence could advertise to insects in the same way that colour and smell indicate to insects that nectar is available in existing plants.
There are squid that release bioluminescent mixtures instead of ink, using light instead of darkness as distraction and cover. A plant could have leaves that glow when disturbed (perhaps initially evolved as a damage response, but then amplified over the generations as the distraction caused is beneficial). It could be of benefit to small prey animals to have a swaying mess of glowing leaves causing distracting cover when it darts into the undergrowth to evade a predator. Although this is not of direct benefit to the plant, being a common escape route for small animals is likely to lead to the regions around such plants being visited more often by both predators and prey, leading to more defecation with the benefit of increasingly fertile soil.
Some animals glow to distinguish themselves as poisonous. Plants already use bright colours to make their fruit distinguishable, so that animals can learn which fruit are safe to eat. Glowing fruit could provide a similar benefit, ensuring that the majority of the fruit are eaten by animals that will survive to spread the seeds. Glowing could also be used to indicate ripeness so that unripe fruit is not eaten prematurely.
Some plants respond to their leaves being eaten by releasing unpleasant tasting chemicals so that the grazing animals will move on to other plants. Glowing could potentially accompany this unpleasant taste, to highlight when the effect has begun so animals can avoid the plant.
Of course all of these warning approaches will only work at night, and as such will likely evolve alongside patterns of colour that perform a similar function during the daylight.
Predators using bioluminescence are found in the deep ocean, but it is not unfeasible to imagine a bioluminescent insectivorous plant. Similarly to the mimicry of glowing insects for pollination purposes, a pitcher plant, venus flytrap or sundew plant could attract insects using light. This could either be by mimicking an insect to attract a mate, or simply using the glow to attract nocturnal insects like a lamp.
The Black Dragonfish lives at ocean depths where red light cannot reach, so red creatures appear black and are practically invisible in the darkness of the depths. By glowing red it can illuminate prey that most other predators cannot see. As an added bonus most creatures at that depth cannot see red light, so the prey is not alerted by the glow. Plants on land live in an environment that allows most wavelengths of light to pass unhindered, and with both day and night light sources that are wide spectrum, so this kind of restriction to one colour is unlikely to give a similar advantage.
General illumination may have benefits, simply in bringing increased animal activity and the various associated benefits discussed above. There is the possibility of developing the ability to glow in a colour only visible to some animals, but the probability will be lower as there is much more overlap in light detection in creatures on land. For example, red berries are easily distinguishable from green unripe berries to creatures that can distinguish red from green (such as insects, birds and humans) but to creatures that cannot distinguish red from green (most mammals) the ripe berries do not stand out. However, even creatures that see red and green as the same colour can still see both - they would detect a glow whether in red or green, they just wouldn't be able to tell them apart. An ultraviolet glow would be completely invisible to some creatures (rather than just being an indistinguishable colour). Most mammals cannot see ultraviolet whereas many birds can.
Being able to produce a glow that only some creatures can see may have benefits for a plant. For example, being able to make damaged leaves glow may attract the attention of birds or frogs that can eat the insect larvae responsible. The glow may even be visible through the translucent skin of the larvae, so that those that eat the defended plant will glow and be highly visible to predators. This effect may be strongest if the glow is a colour that the larvae cannot see, so that they have no warning and stay out in the open not realising they are suddenly much more visible.
The simplest answer would be if it attracted bugs. Many flowers have bright colors or attractive smells to get bees and other insects to pollinate them. Maybe in another world bioluminescents helps the plants get fertilized.
Along the same lines, bioluminescents could repel predators, either they are light sensitive or the bioluminescent chemical reaction is poisonus or destructive to them.
