The phrase "press heavy on" means to apply pressure or weight onto something or someone. There are several synonyms that can be used interchangeably with this phrase. For instance, "bear down on", "weigh down on", and "burden with" are commonly used synonyms for "press heavy on". Other synonyms include "oppress", "crush", "exert pressure on", "overwhelm" and "suffocate with". In all, these synonyms serve the purpose of conveying the same sense of pressing down or weighing heavily on something or someone. Whether in a literal or figurative context, these synonyms capture the essence of applying weight, force or pressure onto something or someone.