Since the answer is not
Could this be
A boil? (Alternate answer: Your Appendix/Appendicitis)
I burn without flame, though you may disagree,
Boils can feel like a burning sensation on your skin, and they can be really painful (Appendicitis can feel like your stomach area is burning)
I understand your pain, which stings worse than a bee.
Really, really painful (Appendicitis is really painful as well)
I rupture easily, even though I can last forever,
You can puncture a boil fairly easily, although if you don't touch them, some may get too big to be drained and last a very long time (The appendix can rupture fairly easily, causing a form of appendicitis; although for some people the appendix can remain intact for their entire lives)
I surcease uneasily, I will affect you whenever.
Boils tend not to just "go away", and from what I've heard, although removing one is easy, it's also painful. If you don't remove it, they can continue being painful for a long time (Also, appendicitis needs surgery to resolve -- it doesn't stop very easily either. It can also affect people whenever, it doesn't have to happen to people of a certain age)
I trap people, who are tied to the past.
Boils that happened a long time ago trap people with the pain (Appendicitis is a "trap", because the appendix is a vestigial organ which ties modern humans to earlier humans/primates who needed it to digest raw meat)
The second letter of each line spells "BURST".....this could refer to bursting a boil or the appendix bursting.