Learn French: L'elfe a la potion, mais il oublie qui soigner. - The elf has the potion, but he forgets who to heal.

This sentence uses an indirect question: a question word (qui) followed by an infinitive (soigner). Just like in English ('who to heal'), you can use the infinitive when the person doing the forgetting is the same person doing the healing.

I thought qui was only for subjects. Can it act as the object here ('whom to heal')?

Yes! When referring to people, qui can be used as a direct object ('whom') in an indirect question like this one. If the elf didn't know what to heal (a thing), you would use quoi instead (il oublie quoi soigner).

Doesn't the verb oublier normally need the preposition de?

You are right that to say 'he forgets to heal', you would normally need de (il oublie de soigner). However, when you insert a question word like qui, où (where), or comment (how) to form an indirect question, you drop that preposition.

Could I use guérir instead of soigner for 'to heal'?

You could, but they have slightly different meanings. Soigner means to treat, care for, or nurse someone, focusing on the medical effort. Guérir means to cure them or make them fully better, focusing on the final result.