Burnt

I would say "I burned myself on my motorbike's exhaust pipe."

"I got burnt by my motorcycle's exhaust pipe" sounds just fine to me.

I might sprinkle some expletives about as well. :)

I prefer (like Mara)

I burned my finger on my bike's exhaust pipe.

to

I got burnt by my bike's exhaust pipe

or even

My bike's exhaust pipe burnt me

Three reasons for this, I think. Firstly "I got burnt" has various idiomatic usages, like "I got burnt
by the Nigerian scam". Secondly, it is bad to
make the exhaust pipe seem like a volitional
agent (i.e. something that deliberately sets
out to burn you). I can imagine a comedian
telling a story that relies on the fact that the
bike really is out to get him, but it is not
normal. Finally, "I got burnt" slightly encourages
the reading in which after we are done there is
nothing left but ashes.

That happened to me once. I was so stupid, at a stop light, absent-mindedly reaching for a piece of string that was hanging on the front, from the fabric bike cover I had taken off earlier. Inside of my left forearm, a nice oval burn. Being such an expert, I would either say: I burned myself on the exhaust pipe; or, I got burned by the exhaust pipe. I'd say the second option doesn't really own up to the goof mistake, though. It's very passive, like you were just standing there and along comes this pipe and burns you. Be a man and take the blame. Use the first option and insert the expletives Ingis recommends in front of "exhaust pipe", since that is what they modify.

like everyone else says, 'i got burnt' doesn't really work. Because the original active sentence should be reflexive - 'i burnt myself (on the exhaust pipe)' not 'the exhaust pipe burnt me'. The passive form would therefore be 'i got burnt by myself (on the exhaust pipe)' which is clearly ridiculous.

Using get+passive should be reserved for situations where you are the unfortunate victim of someone/thing else's actions, not your own clumsiness.

I prefer to not to burn myself

I may be wrong, but isn't "burnt" used as an adjective to describe something? I think the correct usage would be "I got BURNED.. etc."