Canonically, bacterial plasmids will not replicate in eukaryotic systems, due to a lack of the necessary control (regulatory) elements which will lead to expression.
It is possible though to construct shuttle vectors that can propagate in prokaryotes and eukaryotic cells (e.g. yeast or mammalian cells like the immortal HeLa cervical cancer-derived cell line).
What you are specifically asking is whether horizontal (lateral) gene transfer between prokaryotes and eukaryotes occurs, and whether it has an effect on the eukaryotic cell. This 2016 paper is an excellent and recent article discussing this question.
Can a plasmid cause cancer?
It has never been demonstrated, to my knowledge.
Is it possible for plasmids to get into eukaryotic cells presents in
the digestive tract?
Very few bacteria can penetrate a healthy intestinal lining. However, in many disorders of the gastrointestinal system, such as in inflammatory bowel disorders, the intestinal lining is compromised, which certainly allows the passage of prokaryotes and/or DNA into human circulation. However, a functional transfer of plasmids from prokaryotes to eukaryotic cells is not demonstrated. Whether the hypothetical transfer would lead to carcinogenesis is also not demonstrated.