There appear to be multiple ways of reading this sentence, depending on how you interpret the context of "rest" and "silence".
Hamlet has been experiencing a great deal of upset and distress during the course of the play. Enough to drive him to madness. So one possible reading of this sentence is that the "silence" of death will finally allow him to "rest". "Going to your rest" is, of course, a common euphemism for death.
It is also the end of the play, as well as the end of Hamlet. "The rest is silence" can be read literally in the context of the performance: there is little left to watch, after which the characters will be silent. And in the context of the play itself, Hamlet himself will be silenced by death.
However, it can also be read as a continuation of the speech and little to do with death itself. Before expiring, Hamlet is talking about the possible outcome of Fortinbras's election. "The rest is silence" could be interpreted as a statement saying Hamlet would vote for him, but the outcome is otherwise uncertain. "The rest" of the possible outcome is "silent": one can imagine a personification of fate or prophecy refusing to speak more on the matter.
If you like puns, and Shakespeare seems to have done, there's one here. Shortly before his dying speech, Hamlet personifies Death and refers to the act of dying as an "arrest". So here he is saying "th'[e/a]rest [i.e. dying] is silence".
Finally, a major theme of the play is mortality and the question of what comes after. We see this in Hamlet's statements about the ghost of his father, about Yorick, about men going to die in battle. We see it, most famously in "to be or not to be" and in an "undiscover'd country". "The rest is silence" could be a statement of uncertainty: Hamlet is about to find out the answer to this question but he cannot tell anyone what it is. In fact, the answer may be "silence", i.e. oblivion. This is your theological connotation.
So the literary value in this short line appears to be one of substantial ambiguity. Which is fitting, given the questions of uncertainty and faith at the heart of the play.