When you say "plant", I think it depends on what you mean. In an Earth context, a plant seems to be an organism evolved to live as part of the environment here, which is a symbiosis of many life forms in the conditions that have existed here. Some plants (e.g. Kudzu) may tend to spread very aggressively and smother other plants to an unusual degree, but every organism on Earth ultimately requires conditions which come from many other species, and relatively stable climate conditions. Every species on Earth which is "too successful" at dominating its environment, tends to eradicate the conditions which allow it to live, resulting in a serious die-off and possible extinction. The planet as a whole however can be said to be alive as a whole, and has much more potential for long-term survival.
In order for one species to be able to take over and dominate an entire planet, it would need to be able to somehow generate its own requisites for continued existence. The easiest way to accomplish that, is not to eradicate the other species, but to encourage and aid other species, which in turn have their own survival requirements and needed conditions to keep in balance. When different species interact to support each other, it's called a symbiotic relationship. Even predator/prey relationships tend to be ultimately symbiotic, as they help control the imbalance of different species' population levels, avoiding toxic conditions, starvation, die-offs and extinctions. This tends to be best supported by a naturally-evolved ecology with plenty of biodiversity, perhaps best exemplified by the Amazon rainforest before foolish humans started destroying it for low-grade cattle pasture.
A planet-consuming "plant" would also need something to cause it to organize itself to eradicate other species and generate its own conditions in some other way, which on Earth or another natural planet, I don't see a likely reason for without something like deranged human thinking at play.
You also describe it as one planet-sized plant, which is a challenge in itself. One species with many members is one thing, but requiring it to be a single member ... well it depends on what you mean. I think it could work as long as much smaller pieces could function independently. To achieve the self-symbiosis, I imagine it would need to have many different forms or components to create its own life cycles.
At which point, I think you're falling back towards something more like life on Earth anyway. Until relatively recently over the span of human existence, most humans have considered themselves to be a part of our living world as a a whole, and not as a separate thing struggling against nature. The wisest answer may be that we are already on a planet-size life-form or even plant, at a macro level. We're all part of the organism, and the idea that we're not, and our nastier industrial activities, are a disease which we need to outgrow to avoid our own die-off.
Back to a more literal vision of your question, another aspect is that if you start with an Earth-like planet and an Earth-plant-like "plant" (say, genetically modified by the evil Doctor Monsanto to intentionally devour the entire planet), there would be challenges to the literal physical task in that only the topsoil and oceans of the planet are very useful nutritionally. If evil Doctor Monsanto intends his plant to devour the very planet beneath the outer layers, he needs to find something that will do something to inorganic rock, and then to magma, molten iron, etc...