Most of the substantive stuff in Bultmann's thought seem closest to Levinas, who, like Bultmann, is a thinker of the Radical Other. For both, the source of ethics and religion is the call of the Other. In other words, I don't act ethically toward you because you're similar to me; I act ethically because of your difference from me. This notion underpins Bultmann's theory of the "kerygma" of the text, which he defines as an existential relation to the text, which calls to me (to simplify: it resonates in my soul. The Bible isn't a set of propositions which are true or false, but something that calls to me from a distance and stirs me).Bultmann's great work, actually, is not Jesus Christ and Mythology, which he wrote trying to explain his "demythologizing" theology to the laity; his great work is his magisterial study of The Gospel of John. It's true Bultmann was friends with Heidegger (they were professors together at the time), but he read John's gospel side by side with Kierkegaard's Philosophical Fragments. The existentialism is much more Kierkegaard's than Heidegger's, who is, in fact, more of a phenomenologist anyway (the difference is hair-splitting, actually), and where Bultmann can't avoid it, he includes long footnotes quoting Kierkegaard's pseudonymous work (so it is, and it isn't, Kierkegaard's, at one and the same time. Paradoxes abound).
For what it's worth, merging Levinas and Christian theology got kind of hot in the 80s, so a resurgence of Bultmann, read through Levinas, is possible and would be kind of neat to see. That would require reading him against the grain of the conventional wisdom, but it's definately do-able.
The idea of charging theology by reading Bultmann through Levinas is a profound one, for the reasons JPE gives here. Conventional wisdom in ecclesiastical circles today is all about "community," but "community" is all about the Other. And that, as Derrida says, puts us face to face with the mysterium tremendums, that which always makes us tremble. It is, in fact, an existential trembling, a terror in one's being, a fear of eradication. Which all springs from Romanticism, actually, which will always be with us, now. As Kierkegaard understood, his philosophy and theology being wholly a product of Romanticism.
Much to do, and so little time to do it all....