Well the infinitive works in this sentence simply as its basic function and not in an idiomatic way. Translated directly to english it reads as "I will not give your body to burn." and "Give to her to clean your wounds.". Both of the infinitives work as the infinitive should, which is to compliment or modify the other verb used in the phrase (the non-infinitive verbs being "vazhok" and "azhas").
Right, but you wouldn't translate those sentences as "I will not give your body to burn." and "Give to her to clean your wounds.", You would translate them as "I will not allow your body to burn.", "Let her clean your wounds." because "to give" is not used to mean "let" in English. What I mean when I say "azhat" and "to have" are being used idiomatically is that, to a non-native speaker, their secondary meanings of "to let, allow" and "must" are not obvious from their primary meaning. It would not be obvious for a monolingual French speaker learning English, for example, that "to have" can mean "must" since "avoir" in french can't be used that way, so a sentence like "I have to eat" would look very odd to them and they would either have to be taught this idiomatic use or infer it from context. If I were a monolingual Spanish speaker learning Dothraki and ran across the sentence "Me azha morea adakhat" and didn't know "azhat"'s secondary meaning, I might incorrectly guess it means "Él les da de comer" - "He gives them (something) to eat.", because the verb "dar" can't be used to express permission in Spanish, it's an idiomatic usage particular to Dothraki.
the "to" in the sentence is not related to the verb "to eat". It is apart of the auxillary verb "have to", which is a synonym of the other auxillary verb "must". The sentence given wouldn't translate to "Anha mra qora adakhat"; it would translate to "Anha'th adakhak".
Well, I'd argue the "to" in "to have to" is simply a remnant of a missing infinitive to signal that the verb is being used to express obligation, but eh, doesn't really matter, my main point was as explained above.
The only idiom in those lines of dialogue is "Azhas maan", which would tranlsate to "Allow her". Coincidentally, this could also be expressed using the Jussive (but is significantly muddier), which would be "Affisates me zis yeri".
I don't know, I guess there's just a subtle difference between "azhat" - sp. "dar; dejar", fr. "donner; laisser" and the jussive, sp. & fr. subjunctive in the main clause, both English "let", that we sense. The jussive, as its etymology suggests, just sounds a tad more commanding for us.