What is the crime of lèse-Bourbaki?

I don't think it was honoring Bourbaki, but rather gently mocking them.

As stated in the comments and other answers, the Bourbaki group was known for its lack of redundancy : if you read the books, you see they never repeat definitions or arguments, instead they always refer to previously written books with a concise and cold reference (I don't have the format in mind but it'll be something like [Book 5, Ch.6, S.5, §4.3])

Lang is saying he will not do this and he coined the term lèse-Bourbaki. In doing so (if I'm not mistaken, he's a French speaker, and so he's not making a mistake in the use of this image) he's not equating them with royalty, but rather implying that they see themselves as such or that others see them as such: he's gently mocking the status of Bourbaki (which was in a sense the status of royalty, back in the days, at least in France).

I'm saying this because, as a native French speaker, I know how and why French speakers use phrases relating to kings and queens: we use them to disqualify people, rather than honor them. When a child throws a tantrum because s.he is denied something s.he wanted, it's not rare (of course you don't hear it all the time) to hear his/her parents refer to their child as a king or queen, and tell him/her jokingly that they were just victim of a "crime de lèse-majesté".

As a French speaker, this is what I understand from this quote (and again, Lang was himself a French speaker if I'm not mistaken): it's not as bad as a child's tantrum but you can certainly imagine Lang with a wry smile while writing this. So to answer your question in the comments, it's definitely related to "lèse-majesté", but it's very likely not to honor Bourbaki.

(Note that the habit of using phrases and sentences relating to kings/queens mostly as derogative is very common in France and is probably a heritage of our many revolutions and continued failures to attain a democracy)

Bourbaki notoriously references old theorems in a book in order to prove a theorem that might come much later in order to not repeat arguments. Lang is saying that he will try not to do this (he will commit the sin of doing something contrary to what Bourbaki would want.)

This is not really an answer to the stated question, which the other answers address in a quite satisfactory manner, but I would like to point out that Lang somewhat misleadingly describes the Bourbakist principle of economy. Not only should one not repeat definitions or arguments identically, one should more generally not explicitly prove any result (say about vector spaces) if it can be seen as a special case of a result valid in a more general context (say of modules over a not necessarily commutative ring). In that case it should instead just be recalled that the more general result, necessarily appearing at some earlier point due to the "the from general to the specific" ordering of the Bourbaki oeuvre, has a certain implication in the current (more specific) setting.