is/or

I'd use "with" rather than "on". It should be 'cheese or lettuce', I suppose, but since such utterances, dear Yoong Liat, are made verbally in the most informal of situations, namely at the service counter of a burger establishment, it hardly matters, and you could risk saying "I don't want cheese and lettuce with my burger" without fearing punishment by the grammar police!

More correct would be "I want neither cheese nor lettuce with my burger", but people who talk in a bookish fashion like that get funny looks. (I should know, I am one of them)

Many thanks, Contrex, for the detailed and clear explanation.

Best wishes

They mean different things (or imply a different meaning)

If you say "I don't want cheese and lettuce on my burger" it means that cheese & lettuce were an option offered together.

"I don't want cheese or lettuce" means of course that you want neither. This would be the normal choice of words.