Μην ξεχάσεις την απόδειξη, σε περίπτωση που χρειαστεί επιστροφή.

Μην is used for prohibitions/negative commands in Greek (often with the subjunctive).
Δεν is used for negating statements/facts (e.g., Δεν ξεχνάω = I don’t forget / I’m not forgetting), not for telling someone don’t do X.

ξεχάσεις is 2nd person singular (you) in the aorist subjunctive of ξεχνάω/ξεχνώ (to forget).
With Μην + subjunctive, it functions like a negative command: Μην ξεχάσεις… = Don’t forget… (especially “don’t forget this one time / in this situation”).

Yes—often a nuance of one-off vs. ongoing/habitual:

• Μην ξεχάσεις την απόδειξη: Don’t forget the receipt (this time / when you go now).
• Μην ξεχνάς την απόδειξη: Don’t forget the receipt (as a habit / don’t be forgetting it).

Because την απόδειξη is the direct object of the verb (forget what? → the receipt), so it goes in the accusative case.
Dictionary form is η απόδειξη (nominative), but it changes with function in the sentence.

απόδειξη literally means proof/evidence, but in everyday Greek it also means a receipt (proof of purchase). In stores you’ll commonly hear η απόδειξη; = the receipt?

σε περίπτωση που is a fixed-like expression meaning in case (that).

• σε
  • περίπτωση = in (the) case
• που introduces the clause (similar to that/when in English, depending on context).
  It’s a very common way to introduce a contingency.

χρειαστεί is 3rd person singular aorist subjunctive of χρειάζομαι (to need), used impersonally here:
σε περίπτωση που χρειαστεί… = in case it is needed / in case you end up needing…
Greek often uses “it is needed” structures where English uses “you need.”

Because επιστροφή is being used in a general/indefinite sense: in case a return is needed.
You could also see:

• μια επιστροφή = a return (more explicitly indefinite)
• την επιστροφή = the return (more specific, context-dependent)

It can mean several kinds of “return” (coming back, refund/return), but with απόδειξη (receipt) it strongly points to a shop return/refund context.

It separates the main instruction (Μην ξεχάσεις την απόδειξη) from the conditional phrase (σε περίπτωση που…). It’s the same kind of pause you’d naturally make in English: Don’t forget the receipt, in case…

Change the verb to match the person/number:

• Singular informal: Μην ξεχάσεις…
• Plural / polite (to one person): Μην ξεχάσετε…
  So: Μην ξεχάσετε την απόδειξη, σε περίπτωση που χρειαστεί επιστροφή.

A few common ones:

• απόδειξη is pronounced roughly a-PO-dhik-si (the ει sounds like ee, and ξ = ks).
• χρειαστεί: hree-a-STEE (stress on the last syllable: -στεί).
• Μην: final -ν is usually pronounced before a following consonant here (Μην ξε-…).