I can't think of any reason why plants wouldn't be able to get cancer, but I've never heard of a plant growing a tumor. I've also never seen a plant with a noticeable abnormal growth. Can plants get cancer?
Yes, plants of all sizes can have cancerous growths.
Agrobacterium tumifaciens, the causative agent of crown gall disease, produces what is called a tumor. See this link for detailed information on these growths. Alternatively, use a plant physiology textbook to look up the above terms. (Here, is where a textbook is better than a single abstract in PubMed.)
I am certain that you have seen such growths on trees, such as this. Other smaller versions can occur on smaller plants - on stems and leaves, for example. The plant builds up tissue around the A. tumifaciens infection in an effort to isolate and contain the infection and its effects.
Burls/Burrs are often defined as tree tumours, and as analogous to cancer in animals. There seems to be numerous reasons why burls are formed (more research is needed on the topic), which include wounds, insects, fungi, DNA changes, environmental stress and viruses. However, burls are localized and often relatively benign, and since plant cells do not move around they cannot metastasize (see e.g. "Can Plants Get Cancer?" from the New York Times). Since burls have multipe causes the 'Cancer/tumour' label might not fit very well to all types of burls, and in some cases normal shoots can sprout from burls that have been removed from the tree (indicating that the burl was not caused by damaged DNA).
Galls are related tumorous growths that are usually caused by parasites (insects, virus, bacteria, fungi), and are often induced by plant hormones (at least from insects/mites, see Insect and mite galls). However, in the case of crown gall caused by the bacteria Agrobacterium tumifaciens the tumorous growth is due to a DNA transfer, which will permanently modify the infected cells (Gheysen et al 1987).
As you can see, the answer to your question depends on if you are interested in tumorous growths in general (where many types of burls and galls should qualify) or cancer in particular (if defined as unregulated cell growth caused by permanent DNA damage).
Could not fit in a comment. The following is purely intuitive, I have not much knowledge in this field and no reference. I am just guessing and I'd welcome some corrections. I was thinking about three points which effect opposite each other.
1) In plants, there are no cells specialized in transporting stuff throughout the body (I think!) while there are many cells that travels a lot in animals. Therefore, I0d guess that plants cancer are much more likely to only remain localized while animal cancer are more likely to create metastasis. As a consequence, I'd expect that cancer would be more lethal in animals than plants.
2) The number of mitosis occurring in the lifetime of an individual is probably positively correlated with the occurrence of cancers. Because plants tend to be bigger than animals, I would expect more mitosis in plants and therefore, I would expect that cancer are more common in plants than in animals.
3) According to life history theory, the intensity of selection on a gene decreases as the ratio of the fitness pre- and post- expression of the gene. Because plants never really age and stop reproducing, I would expect that selection against cancer would be stronger in plants than in animals resulting in a less cancer in plants than in animals.