Heat stress is an important environmental factor affecting livestock production worldwide. Primary hepatocytes and preadipocytes derived from Hu sheep were used to establish a heat stress model. Quantitative reverse transcriptase-PCR (qRT-PCR) analysis showed that heat induction significantly increased the expression levels of heat stress protein (HSP) genes and the N6-methyladenosine (m6A) methylation modification genes: methyltransferase-like protein 3 (METTL3), methyltransferase-like protein 14 (METTL14), and fat mass and obesity associated protein (FTO). Heat stress simultaneously promoted cell apoptosis. Transcriptome sequencing identified 3980 upregulated genes and 2420 downregulated genes related to heat stress. A pathway enrichment analysis of these genes revealed significant enrichment in fatty acid biosynthesis, degradation, and the PI3K-Akt and peroxisome proliferator-activated receptor (PPAR) signaling pathways. Overexpression of METTL3 in primary hepatocytes led to significant downregulation of HSP60, HSP70, and HSP110, and significantly increased mRNA m6A methylation; FTO interference generated the opposite results. Primary adipocytes showed similar results. Transcriptome analysis of cells under METTL3 (or FTO) inference and overexpression revealed differentially expressed genes enriched in the mitogen-activated protein kinase (MAPK) signaling pathways, as well as the PI3K-Akt and Ras signaling pathways. We speculate that METTL3 may increase the level of m6A methylation to inhibit fat deposition and/or inhibit the expression of HSP genes to enhance the body’s resistance to heat stress, while the FTO gene generated the opposite molecular mechanism. This study provides a scientific basis and theoretical support for sheep feeding and management practices during heat stress.