People can burn things, or things can burn in a fire, or fires can burn.
"After you read this message, please burn it."
"I watched the logs burning for several hours."
"The fire burned for several hours."
If you "ignite" something, you add the spark that makes it start burning. People can ignite things or things can ignite.
"I ignited the fire."=I started the fire
"After I prepared a pile of tinder and kindling, I ignited it."
"The spark caused the pile of gunpowder to ignite."
"Inflame" is almost always used metaphorically, for love and similar emotions or hate and similar emotions.
"Jill's dress inflamed Bob's desire."
"The nation's new policies further inflamed the hatred of its enemies."