Getting fired definitely happens during psychiatry residency. At all of the 13 places I interviewed at (mid and low tier programs) I made sure to ask the chief resident how many residents had been let go or terminated in recent memory. Eight of the thirteen programs had lost one resident in recent memory. Three had been let go for issues with repeated tardiness, absenteeism, lying about not showing up to rotations, etc. One had been let go for engaging in a sexual relationship with a therapy patient. One had been let go for substance abuse issues. Two had been let go for not passing step 3 by the program's deadline (seemed like they would be re-hired in a year if they passed though). One had been let go for failure to perform as expected. This last program lost an additional resident because said resident left voluntarily as he "didn't want to practice medicine anymore."
So yes, getting fired does happen during psychiatry residency. It's a job after all. Uprofessional behavior seems to be the biggest factor in one getting terminated, so as long as you're "professional," which largely means that you show up, finish your work, don't lie, and don't engage in sexual misconduct or drug misuse, you'll likely be okay.
Keep in mind that I asked the senior about residents who had been lost in "recent memory", which likely means within the last four or even five years. The majority of places I interviewed at had 4-6 residents per year....so based on this, the termination rate among the programs I interviewed at is something like 2%. Also keep in mind that I interviewed at low and mid tier programs. IMG's make up a larger percentage of residents in these programs, and unprofessional behavior during one's career has been associated with graduating from a foreign medical school.
