Infection is one aspect. Another aspect is the battery system that powers the implant, they're usually implanted in the chest with a wire that goes up the neck to the skull. There was a case[1] where tension developed over that wire, leading to pain, immobility and destruction of the connection in brain tissue:
> Steve had the surgery at Stanford, in November, 2012. After the surgery, he had “severe cognitive decline” and a slew of physiological adversities. “The leads [wires] were 18 inches longer than they needed to be, so they coiled it up in the chest and at the top of the head; I could feel them externally,” he says. “And the leads were too tight. I could move my ear and my chest would move, too,” he says of a condition called “bowstringing,” whereby scar tissue encapsulates the wires (partly from the body’s natural response to foreign material), which has been documented in DBS cases and can cause permanent complications. Steve also had many symptoms that were ultimately diagnosed as shoulder and jaw muscle atrophy, spinal accessory nerve palsy and occipital nerve palsy. He reported all adverse effects immediately and continuously throughout the first year of the study, but the trial doctors continually told him that they’d never heard of such symptoms with DBS, even though nerve damage and DBS wire-related “hardware” complications were among the potential risks listed on the informed consent document.
Because they're relatively experimental, it's almost impossible to find a doctor/surgeon/etc that will choose to work on you should you run into complications. If you have problems with the programming of the devices themselves, there isn't much you can do as a patient, and even specialists can't help you. The only people who can help you are those who developed the device. That can be a problem for a device that's meant to remain implanted until death. Removal is also a huge issue, because brain tissues grow on and around the implants. There are people who want their implants removed, but can't find a doctor who is willing to remove them because of the potential for brain injury and the resulting liability.
They can also cause personality changes, suicidal behavior and even homicidal behavior[4]. There are documented cases of increased impulsivity and impaired executive function, which have led to pathological gambling and shopping. Breakdowns of relationships and the ability to work are also documented[3].
Here's an article on the subject[1], and there are numerous studies that look into such effects, like this one[2] that aggregates dozens of studies.
