The substantial personal, societal, and economic impacts of opioid addiction drive research investigating how opioid addiction affects the brain, and whether therapies attenuate addiction-related metrics of brain function. Evaluating the connectivity between brain regions is a useful approach to characterise the effects of opioid addiction on the brain. This work is a systematic narrative review of studies investigating the effect of abstinence or interventions on connectivity in people who are dependent on heroin (HD) and healthy controls (HC). We found that HD typically showed weaker connectivity than HC between three functional networks: the Executive Control Network, Default Mode Network, and the Salience Network. Abstinence and Transcranial Magnetic Stimulation (TMS) both attenuated differences in connectivity between HD and HC, often by strengthening connectivity in HD. We observed that increased connectivity due to abstinence or TMS consistently related to decreased craving/risk of relapse. Using these findings, we present an "urge and action framework" relating therapeutic factors contributing to craving/relapse, connectivity results, and neurobiological models of HD. To inform future research, we critically assessed the impact of study design and analysis methods on study results. We conclude that the weaker between-network connectivity in HD and HC and its relationship to craving/relapse merits further exploration as a biomarker and target for therapeutic interventions.