Learn Spanish (Latin America): Tu perro nunca debe lamer su herida.

While Spanish usually avoids possessive adjectives for body parts in favor of a reflexive pronoun plus a definite article (like lamerse la pata for licking a paw), a wound (herida) is a temporary affliction, not an inherent body part. Because of this, it is very common to just use the possessive su herida. That said, lamerse la herida is also perfectly natural and widely used!

I always get confused between deber and deber de. Why is it just debe here?

Deber without de expresses strong obligation or necessity—something you must do (or in this case, must not do). Deber de expresses an assumption or probability, like saying something must be true. Since you are giving a strict medical instruction about the dog, you use the obligation form: debe.

I thought Spanish required double negatives. Why isn't there a no in this sentence?

Spanish only requires a double negative if the verb comes before the negative word. Because nunca is placed directly in front of the verb (nunca debe), you do not need no. If you wanted to put nunca at the end of the sentence, you would need the double negative: Tu perro no debe lamer su herida nunca.