This is really more a question of English usage than it is a question of translation. All three words are potentially adequate, semantically speaking, as translations of el olvido. The question is more one of connotation, suggestion, or register in English than it is one of meaning.
Oblivion in English is a fairly formal or poetic word. It is much less likely than the other words to be used in ordinary speech, and is more commonly used in educated phrases, in ceremony or in evocative writing. Etymologically speaking, it is ultimately a cognate of el olvido (both are derived from ancient Latin oblīvīscor, "I forget"). Its evocative, formal-register features might make it more desirable to use in the context of a proverbial phrase like this one. On the other hand, oblivion also has very strong connotative connections with death, sleep or unconscious not simply with the loss or letting-go of memories in active waking life. (The few common English phrases which make use of "oblivion" are specifically contexts that emphasize this connection, e.g. "drank himself into oblivion," etc.) That would to make an English proverb suggesting that oblivion is one of the forces that helps us to live kind of strongly paradoxical. It's OK for proverbs to say things that seem paradoxical, but that is a sense of paradox that connecting el olvido with ayudar a vivir in Spanish does not really carry.
Forgetfulness and forgetting are more commonplace English words which are used very frequently. Forgetfulness often describes a character trait of people (i.e., being prone to forget things; sometimes but not always this is used to indicate that their disposition to forget is a flaw or a vice), although it can also be used to describe the process of a memory being forgotten or the state of a person who is forgetting something in particular (as opposed to a general disposition).
Forgetting used as a noun describes the process or activity of losing a memory or letting go of one; in English proverbs it is often connected morally with the process of forgiveness or grace. ("Forgiving and forgetting" or "forgive and forget" are very frequently used as a proverbial pair, the direct equivalent of Spanish perdonar y olvidar.) It has no strong connections with death, sleep or unconsciousness; if anything it is strongly connected with the continuation of life and ongoing activity (for example in Shakespeare, Juliet's "... I would forget it fain, / But, O, it presses to my memory / Like damnèd guilty deeds to sinners' mind ..." Act III, Scene 2, etc.) A downside relative to either "oblivion" or "forgetfulness" is that "forgetting" can describe an active process, but it often indicates something more like a passive lapse of memory, and possibly a temporary one (e.g., I might be forgetting my friend's phone number for a moment, only to recall it later), which counts against the ability to understand it as a powerful active force or capacity within someone.
Nevertheless, speaking for myself, as a native (American) English speaker, in most contexts I would translate el olvido here with "forgetting" or something closely connected with it, unless the context strongly called for me either (1) to adopt a much higher or more ceremonial register of speech; or (2) I actively wanted to embrace the paradoxical sound of connecting help us to live with oblivion.
As a side note, concerning a word you didn't ask about: English force is directly cognate with Spanish fuerza and is typically fine as a translation, including here. But as a close judgment call here, I'd say that it is somewhat more common in Spanish to use fuerzas to describe inner capacities or powers that a person has in their will, mind or spirit, whereas in English force in these contexts would most commonly be used to describe external forces that constrain or act contrary to a person's will, mind or spirit. It would be a bit more ordinary to describe what a person has within them as "powers" or "capacities."
If you want to strictly preserve original word order: "We have two powers [forces] that help us to live: forgetting and hope."
If you want to preserve something of the rhythm of the line (given that "hope" is a much shorter word than "forgetting"): "We have two powers [forces] that help us to live: hope and forgetting."