Psychedelic drugs aren't what most would consider addictive. However, there can be longterm effects from taking them, especially if you have a 'bad trip.' The thing about these drugs is the fact that they change the brain's chemistry and alter neuron paths.
If psychedelics are used for an extended period of time, it is possible to have flashbacks. Still worse, the persistent use of these drugs can change the brain so much that mental illness is developed. Illnesses that can be developed include psychosis, bipolar disorder, depression, and anxiety.
The long term side effects of taking psychedelics are more related to indirect effects that actual physiological symptoms of long term use. Psychedelics, unlike opiates, are not typically addictive, although some people can be addicted to the feelings and sensations of psychedelic use.
Long term use of psychedelics can have an impact on your brain patterns and ways of thinking. It can cause you to have a different perspective and maybe value things that you would not have valued had you not started using psychedelics. However some people do develop pyschosis or anxiety with long term use of pyschedelics.