Schistosomiasis is a poverty-related chronic disease that affects over 240 million people across 78 countries worldwide. In order to control the disease, the World Health Organization (WHO) recommends the drug praziquantel against all forms of schistosomiasis. Mass Drug Administration (MDA) programs with praziquantel are successful on the short-term as they reduce the prevalence and infection intensity after treatment, and thus instantly relieve the patient from the burden of its disease. However, epidemiological and genetic studies suggest that current school-based interventions may have little or no long-term impact on parasite transmission. Here, we adopt a Darwinian approach to understand the impact of MDA on the neutral evolution of Schistosoma parasites and assess its potential to eliminate schistosomiasis. We develop a finite island model to simulate the impact of repeated treatments on the genetic diversity of schistosome populations locally (within each host, i.e. infrapopulation) and regionally (within all hosts combined, i.e. component population). We show that repeated treatments induced strong and lasting declines in parasite infrapopulation sizes, resulting in concomitant genetic bottlenecks within the treated individuals. However, parasite genetic diversity recovered quickly in a few generations due to re-infection, and there was little or no impact of treatment on the genetic diversity of the component population when treatment coverage was 95% or lower. This was mainly due to parasite infrapopulations of the untreated host individuals that acted as reservoirs of genetic diversity, sustaining the diversity of the component population. Hence, lasting declines in parasite genetic diversity were only observed when coverage of treatment was 100%, resulting in population crashes after a minimum of six treatment rounds. We argue that achieving a full coverage of treatment is highly challenging for most endemic regions in sub-Saharan Africa, and conclude that MDA alone has little potential to achieve elimination within a conceivable time frame. Our results raise skepticism about the current WHO goals of elimination of schistosomiasis by 2025.