I recently started to have a heated/burning sensation under my big toe. No associated pain noted. What could cause this?

Few things. Could cause this. First, i would examine the skin to see if there is any sign of cracked skin, blistering or scaling as a fissure or athletes foot may be present. If not, i would question you with regards to low back pain or if you are possibly diabetic. I would also evaluate you for compression neuropathy of the deep peroneal nerve or tarsal tunnel syndrome.
Nerve entrapment. Burning pain associated in the foot is mostly from nerve entrapment syndrome. This is caused by over-pronation of the foot and thus causes instability across the toes. This commonly may push on the plantar nerves that run between the toes, causing a burning sensation. If it feels better to rub them when it happens, most likely a nerve issue.
Nerve pain. You could also have a tear in the tendon. Go see your podiatrist.