In the sixth stanza of the poem Longfellow is attempting to convince the reader of what the correct attitude towards the past and the future should be. Instead of hankering after the past, as we so often do, instead of being nostalgic about it, we should leave it alone. The past must remain where it is. That's what the speaker means by "Let the dead Past bury its dead!" There's no point dwelling on the past; the time is always now. Life is to be lived in the present, nowhere else.

The same applies to the future. It's all too common for people to concern themselves with what will happen in the years to come. Yet this distracts us from what the speaker refers to as the "living Present". (Note the capitalization of the word "present" indicating its importance). For it is only in the living present that life is to be lived.