"Intents and purposes" versus "intensive purposes"

It's most likely a slurring of the original phrase, but "for all intensive purposes" does make it clear that only the most serious purposes are being considered.

I would probably classify it as an eggcorn.

The original idiom is "intents and purposes." Intents and Purposes are both nouns in this case. With "intensive purposes" intensive is an adjective that describes the intensity of the noun purpose.

"This chainsaw is a suitable tool for the intensive purpose of cutting down this tree."

vs.

"This chainsaw serves our purpose, and is a suitable tool to carry out our intention of cutting down this tree."

Does it serve your purposes and satisfy your intents? Or does it satisfy your intensive purposes?

"For all intensive purposes" is wrong. It was first written by someone who heard the phrase "for all intents and purposes" incorrectly. That is, it's something people who have misheard "for all intents and purposes" but who have never seen it written, have started using. It's meaning - to the extent it has any meaning - would be nearly opposite of the meaning of "for all intents and purposes." For all intents and purposes means, basically, "amounts to." For all intents and purposes, they're married, means that they may not have a marriage certificate, but they behave and live as a married couple. "Intensive purposes," to the extent it makes any sense at all, would be limited to a subset of purposes, i.e. those that are most intensive (whatever that might mean) rather than for "all" purposes and all applications, i.e. intents.