Antibiotic treatment significantly impacts the human gut microbiota, but quantitative understanding of how antibiotics affect community diversity is lacking. Here, we build on classical ecological models of resource competition to investigate community responses to species-specific death rates, as induced by antibiotic activity or other growth-inhibiting factors such as bacteriophages. Our analyses highlight the complex dependence of species coexistence that can arise from the interplay of resource competition and antibiotic activity, independent of other biological mechanisms. In particular, we identify resource competition structures that cause richness to depend on the order of sequential application of antibiotics (non-transitivity), and the emergence of synergistic and antagonistic effects under simultaneous application of multiple antibiotics (non-additivity). These complex behaviors can be prevalent, especially when generalist consumers are targeted. Communities can be prone to either synergism or antagonism, but typically not both, and antagonism is more common. Furthermore, we identify a striking overlap in competition structures that lead to non-transitivity during antibiotic sequences and those that lead to non-additivity during antibiotic combination. In sum, our results establish a broadly applicable framework for predicting microbial community dynamics under deleterious perturbations.