Elevated vitamin D levels increase resting metabolic rate independently of changes in physical activity or food intake.

Study: Hypothesized pathways for the association of vitamin D status and insulin sensitivity with resting energy expenditure: a cross sectional mediation analysis in Australian adults of European ancestry

This study found that people with higher vitamin D levels tended to burn more calories at rest, even when other factors like insulin levels were taken into account — suggesting vitamin D itself may boost your body’s resting energy use.

Study: Vitamin D Status and Resting Metabolic Rate May Modify through Expression of Vitamin D Receptor and Peroxisome Proliferator-Activated Receptor Gamma Coactivator-1 Alpha Gene in Overweight and Obese Adults

This study found that people with higher vitamin D levels tended to burn more calories at rest, even when their body size was taken into account, and that a gene linked to vitamin D (VDR) might be why. It doesn't prove taking more vitamin D will boost metabolism, but it strongly suggests a connection.

Study: Effects of vitamin D on primary human skeletal muscle cell proliferation, differentiation, protein synthesis and bioenergetics.

This study found that vitamin D makes muscle cells burn more energy, even when they're just sitting there. That’s exactly what the claim says vitamin D does—boosts your body’s baseline energy use without changing exercise or eating.