They're puns pretending to be similes, which is part of the source of the humour.
"Make like" is an American English expression that can mean "behave as though" ("He keeps making like he's the smartest guy in the class, though he's the only one that keeps failing") or "behave in the manner of" ("He camped it up and made like Diana Ross").
For that reason, "Make like an X" sets us up for a simile or analogy. At "I'm going to make like a banana" we're expecting for the next thing to be something a banana does, that the speaker is claiming he is going to likewise do. Already this promises a punchline because there isn't that many things bananas are notable for doing, that people can do (though once you're used to the form, "I'm going to make like a banana and slowly ripen before rotting and turning to mush" has a humour of its own).
Then we get "... and split", which is the pun. As a pun it is doing two things at once. One of them is referencing banana splits, and the other is using split in the colloquial sense of leaving a place. This last is of course the real meaning of the expression; you're going to leave.
It's also notable that the other sense of split is a wrong in two ways. Firstly we're using it as a verb, when the banana-related sense is a noun. Secondly, we're using it as a word fully separate from banana when we don't use it this way; we talk of "banana splits" as a compound noun, not of just "splits".
The second "mistake" isn't included in most variants of this joke, but the first one is. It's the same for example in "make like a tree and leave", as trees don't leave they have leaves. This form adds a different word-clash because trees have leaf singular, leaves plural, not leave.
"Make like a prom dress and take off" is a less strong clash — between an intransitive sense of the verb phrase, and a transitive one, and where "prom dress" would be the object, not the subject. It also adds that taking off isn't particularly associated with prom dresses (equally to putting on, and less so than wearing), so makes us think more strongly about taking off a prom dress, which has sexual implications.
From the list Callithumpian gives us, most are close to one these in mechanism. A few depend on specialised knowledge to get, and therefore appeal to the in-group that would understand them, acting as a shibboleth.
Sadly some just don't get it. "Make like an airplane and take off" tries to pun one sense of take off with another sense that's closely related, and based on it. "Make like a jet and zoom" is by someone who just doesn't understand that they are meant to be puns, and they are trying to pun the sense of zoom that means "go really fast" with the sense of zoom that means "go really fast". This is why it's not funny.
The social difference between self-referential use and insulting use is because if directed at someone it changes to the imperative mood, and ask that they leave. This has two implications.
- We have social rules about using the imperative, in which "please" and similar soften it from a command to a request, that it is considered impolite to leave out in many circumstances.
- We have social rules against telling someone to go away, in a lot of situations. Generally if we need someone to leave we will at least explain why, to soften this.
Here you've not just broken those two guidelines to polite behaviour, but you've also made a joke of it. Barring a context that justifies this (friends who are often jokingly insulting each other), it is therefore impolite to the point of insulting.
Edit: I think it's worth looking at the older form that StoneyB found in the same way we have the current form above, because it reveals something else going on in the newer form, that we can't see otherwise.
The older form is simpler, a straightforward pun of hoop in the sense where it is a variant of whoop, and hoop as in a rigid circle, which can hence roll away. It's use of make is different—"make a noise" is sense I probably need not explain—and the pun does not have the "deliberate mistake" level of nonsense of the "banana" and "tree" variants.
It's still got a nice sprinkling of nonsense though, it doesn't just pun in dragging in another sense to colour the main one, but jumps us comically from one to the next. There's a certain forced quality to "make a noise like a hoop" rather than "make a hoop" or "make a hoop noise" (and it's less good to our ears as hoop in that sense is now uncommon at least IME). That somewhat forced quality though is necessary if it's going to both make sure we've the noise sense of hoop in mind, and also jump us straight to the ring sense.
What this adds to my answer above, is that at some point people knew this as a common joke (StoneyB has demonstrated that well), so the more recent form was subverting a well-known joke in using make in a different sense of the word to how that form did.
It's not a vital piece of the joke, so the joke still works today, but it means there was yet another piece of humour at work when the "banana" and "tree" forms were first invented.