There used to be a famous German comedian - Heinz Erhardt - who was well-known for forming such funny sentences (look up his stage performances on YouTube, he was very creative with the language). As has already been pointed out, the form is that of a Zeugma, using a contracted verb in two slightly different meanings:
Ich heiße nicht nur Heinz Erhardt sondern auch Sie herzlich willkommen!
Ich fror vor mich hin, denn nicht nur meine Mutter, auch der Ofen war ausgegangen.
In your example, lassen is contracted from two slightly different forms:
- Etwas außer Acht lassen (to disregard something)
- Jemandem etwas lassen (to leave something to someone)
as you have realized already - which is exactly the joking form that Erhardt used. As you can see from the examples, sentences like that can be formed and are quite well understood, but from the fact that this stylistic form is used by a comedian you can deduce it sounds really funny to a native speaker. So, outside comedy, this is not common at all.
Things are a bit different when the contracted verb has absolutely identical notion for both cases - In this case, nothing is wrong or funny about contracting the two verbs into one. On the contrary, not contracting the verbs would sound unusual.
Er ließ sowohl Kosten als auch Gefahren vollkommen außer Acht.
Nicht nur meine Mutter, sondern auch mein Vater war ausgegangen.