Weigh on is a phrase used to describe a feeling of mental or emotional burden. Similarly, there are many synonyms for the phrase that can be used interchangeably. For instance, the expression "weigh down" can replace weigh on to convey a sense of heavy pressure. Words like "burden," "bother," "trouble," and "oppress" can also effectively replace weigh on. Moreover, "haunt," "prey on," and "plague" can be used to suggest a persistent and distressing mental state. Ultimately, the precise synonym for weigh on will depend on the context of the situation, but there are plenty of options available to convey a sense of emotional or mental distress.