Sometimes helpful to work backwards:
いられない = Can't live, can't exist, can't "be"...
感じずに = without feeling. Kanjiru would be the root verb, but this has the ず ending which negates the verb.
意気に = 意気 means enthusiasm, spirit, motivation, energy, passion, etc...
Actually 意気に感じる is kind of a set phrase meaning "feeling someone else's energy/passion" or "being inspired by someone else's passion/energy". So the negated form (意気に感じず) would be "not feeling, or not being inspired by someone else's energy".
So it would mean "I can't live without feeling (your) energy/enthusiasm" or to turn it around, "Your enthusiasm/energy/passion is what I live for". That is what I think it means, anyway. There could be some widely accepted alternative meaning that I am missing.