There’s good news and bad news here.
The bad news is that our metabolic rate depends on our size, simplistically speaking. Therefore, in order to increase our BMR or Basal Metabolic Rate, we’d have to gain more weight, which defeats the purpose.
The good news is that BMR is one component of our body’s Total Energy Expenditure (TEE). There is at least one component of TEE that can be significantly increased. To learn more, head over to the answer for, How can I increase my BMR?