Aneuploid yeast cells are in a chronic state of proteotoxicity, yet do not constitutively induce the cytosolic unfolded protein response, or heat shock response (HSR) by heat shock factor 1 (Hsf1). Here, we demonstrate that an active environmental stress response (ESR), a hallmark of aneuploidy across different models, suppresses Hsf1 induction in models of single-chromosome gain. Furthermore, engineered activation of the ESR in the absence of stress was sufficient to suppress Hsf1 activation in euploid cells by subsequent heat shock while increasing thermotolerance and blocking formation of heat-induced protein aggregates. Suppression of the ESR in aneuploid cells resulted in longer cell doubling times and decreased viability in the presence of additional proteotoxicity. Last, we show that in euploids, Hsf1 induction by heat shock is curbed by the ESR. Strikingly, we found a similar relationship between the ESR and the HSR using an inducible model of aneuploidy. Our work explains a long-standing paradox in the field and provides new insights into conserved mechanisms of proteostasis with potential relevance to cancers associated with aneuploidy.