The first sentence is grammatically correct: ""In case the problem persists, inform the Front Desk Officer and ask for assistance." The second construction that knocks off the preposition "for" in the phrase "ask for assistance" is awkward and grammatically wrong: "In case the problem persists, inform the Front Desk Officer and ask assistance."
When "ask" functions as an intransitive verb in the sense of making a request, it needs the preposition "for" to link it to an object, in this case "assistance." We must keep in mind that by itself, an intransitive verb can't act on a direct object; it needs a preposition as a go-between. In the particular case of the second sentence you provided, the noun "assistance" becomes what's known as the object of the preposition.
Another way of looking at the problem with the intransitive verb "ask" as used in the second sentence is this: You can't "ask" a concept (in this case "assistance") to do something for you; you ask somebody (an indirect object like, say "Amy") to give you that assistance, as in "Ask Amy for assistance." Keep in mind that because "ask" is an intransitive verb in that sentence, the noun "Amy" isn't its direct object but only an indirect object or intermediary for the verb's action. Neither is the noun "assistance" a direct object in that sentence construction; it is, in fact, only the object of the preposition "for."