What are the suttas in which the Buddha describes suffering as "burning" I remember reading suttas in which the Buddha describes suffering, lust or other unwholesome mind states as burning. What are these suttas?
You may be thinking of the Fire Sermon (SN 35.28).

It says that the senses and so on are aflame ... aflame with lust, hatred, and delusion.

In more detail, it says the following are aflame:


Sense-organ, sense object, sense-consciousness, and sense-contact
Any feeling[s] arising from the senses ... including suffering (dukkha), but also sukha


... and aflame with:


The three poisons
Also birth, ageing, and death
And sorrow and so on