In English we would say the same thing using let "I let my hair cut" where 'cut' is p.p.(past participle). because the hair "is cut". In the sentence "Ich lasse meine Haare schneiden" why isn't schneiden in p.p. form? I didn't learn passive form yet. But shouldn't it be "Ich lasse meine Haare geschneiden."?
First, the German sentence
Ich lasse mir die Haare schneiden.
would be more idiomatic. The mir here is similar to the mir in
Ich wasche mir die Hände. (I wash my hands.)
The structure of the German sentence becomes clearer if we add who is cutting:
Ich lasse ihn mir die Haare schneiden.
I let (or: have) him cut my hair.
Here, cut is also an infinitive and not a past participle in English. (Btw, the past participle of schneiden is geschnitten.)
This also shows why the mir is helpful in the first sentence. Otherwise, one might get the idea that the hair is cutting something. Indeed we could say
Ich lasse meinen Bruder schneiden.
and context would hopefully clarify that the brother is cutting, not being cut. For a clearer example:
Ich lasse meinen Bruder fahren.
I let/have my brother drive.
(The German sentence actually could have also other interpretation like let go, but that does not matter here.)