Preamble: I think this is a valid and on-topic question for this forum. This type of question can seem inflammatory for the reason that the asker doesn't realise they are entering such well-trodden ground. I hope we can maintain a community that is friendly and welcoming enough to patiently answer such questions.
TL;DR Yes if they have some degree of sentience, then they deserve moral consideration. Practically, though, they're the food source that causes the least amount of suffering.
Extended TL;DR: Yes, if there is good evidence of conscious experience, then of course they deserve moral consideration. More practically, if plants do hold moral value, it's definitely not a good idea to eat animals since those animals had to eat a large mass of plant matter to produce a small mass of consumable meat. For example, for every kilo of beef you eat, you're "eating" about 6 kilos of grain. If you ate the grain directly you'd get more energy, more protein, more fibre, more everything (except for a few nutrients like B12). You'd cause far less suffering by eating plants.
If good science does show that plants are capable of conscious experience and suffering, then yes, they most certainly do deserve moral consideration.
Aside: Debating what "conscious experience and suffering" actually means could be within the scope of this question, but I think we can provide a practical/prescriptive answer without getting into that.
First things first: We must agree that consciousness and sentience aren't binary. Instead, varying degrees of consciousness are possible. Here's a terrible diagram:
Obviously this isn't anywhere near accurate or to scale or in proportion - it's just to illustrate that there is a spectrum of moral value. If we care about the moral value of other organisms, but we also want to survive (debates about selfishness aside), then we should try to ensure that our diets have the smallest practical negative moral consequences.
If you disagree that plants have a lower moral value than animals (like cows, chickens, etc.), then I'd have to call your bluff. I can't imagine that anyone actually believes this. If they'd just as easily burn a dog alive as they would a lettuce plant, then I think my convincing them is outside of the scope of this question.
So then the question becomes: How do we achieve the "smallest practical negative moral consequences"? What is considered "practical" by a person depends on their willingness to forego pleasure for the sake of reducing the harm they're doing to others. It also depends on the scientific consensus regarding the sentience of the various food candidates. It's still early days in our scientific understanding of consciousness and sentience.
Moving beyond these questions into more concrete prescriptions of good behaviour requires a lot of utilitarian and scientific ground-work and is probably best answered on a situation-by-situation basis. As an example of the "moral calculus" that we must deal with as we start to move into this territory: A person may justify eating plants by demonstrating that the effort that it would take to switch to and maintain a algae-based diet would detract from their other positive moral endeavours such that they'd thus be able to do less good in the world. Indeed, it may be quite hard to fulfil one's dietary requirements using only algae today, given that they currently don't exist in supermarkets on a large scale. One hundred years ago, this may have been a sane and valid argument for avoiding a vegan diet (pre algal B12 synthesis, for example).
Achieving the "smallest practical negative moral consequences" is a tough one, but since this is a vegetarian/vegan forum, the question is a usually little more simple: "Should we avoid eating animals?" In this case we can prescribe moral behaviour using only the knowledge that plants have a lower moral value than that of the herbivorous animals that we eat. The argument is simple: The animals that we eat must eat several kilos of plants to produce some smaller mass of food (meat, eggs, milk, etc.). The resulting animal product has less energy, less protein, less fibre and, in general, less nutrients than the original plant matter. Thus, if don't filter our food through the herbivores, we're able to get the same amount of energy, protein, etc. and have killed less plants overall. So we can see that there's no need to know the absolute moral value of plants if we just care about practical ethics.
So, in summary: Yes, plants do deserve moral consideration. We should try to avoid assigning moral value purely on the basis of species membership (or kingdom membership in this case). An organism's moral value should depend on its capacity for conscious experience and sentience as estimated via the available scientific data. Following this understanding, we should be careful not to fall into this trap: "I'd still be causing suffering even if I only ate plants. I can't stop all suffering, so I'll just give up altogether and not worry about morality." That's an irrational and immoral position to take. Instead we should attempt to minimise the suffering that is caused by our diets.
Aside: It's also worth noting that many of the plant products that we eat don't require us to kill or harm the plant. Fruits, nuts and many cereals/grains have evolved to work in symbiosis with animals such that the animals get to eat, and the plants get their seeds spread around (and often get a nice head-start be being embedded in nutrient-rich animal dung). Humans are in a species-level symbiotic relationship with the domesticated animals that we consume/use, but unfortunately the individual's well-being isn't increased by this species-level relationship. For example, bile bear farms may increase the number of sun bears in existence, and in that case sun bears would be in a species-level symbiotic relationship with the humans that breed them, but of course the wellness of the individual sun bears is not increased (rather, it is decreased considerably). So we've got to be careful not to conflate the "success" of the species with the wellness of the individual. In this way, the word "symbiosis" can be deceiving (certainly in its colloquial sense).
P.S. The OP's question was simply about whether plants deserve moral consideration. I've answered a broader question surrounding the "plant suffering" topic because it's hard to wrest the question itself from its common (often fallacious) implications.