Is "The Relic" an ode? What kind of ode is it? Could it be a Horatian ode?

An ode is defined as, "A lengthy lyric poem typically of a serious or meditative nature and having an elevated style and formal stanza structure."

An "ode" does not have a single, defined structure or style, as sonnets, limericks, and haiku all have.  Those styles have rigid definitions, but an "ode" is more open, more inclusive.  The only requirements are that it is "lyric," which means that it seems to have a natural rhythmic flow (many...

An ode is defined as, "A lengthy lyric poem typically of a serious or meditative nature and having an elevated style and formal stanza structure."

An "ode" does not have a single, defined structure or style, as sonnets, limericks, and haiku all have.  Those styles have rigid definitions, but an "ode" is more open, more inclusive.  The only requirements are that it is "lyric," which means that it seems to have a natural rhythmic flow (many odes were meant to be sung), and that it is based on a serious or meditative theme.

The poem you name, "The Relic" by John Donne, seems to fit this definition.  Just looking at the poem on the page shows its length and "formal stanza structure."  A bit of quick analysis shows a somewhat irregular, intricate rhyme scheme pattern: AABBCDDCEEE.  This pattern is repeated in each of the three stanzas. 

The serious nature of the poem is evident right from the start: "When my grave is broke up again...."  (What could be more "serious" a topic than something that starts in the grave?)  This is, overall, a love poem, with the message that the speaker's love for his wife/girlfriend/partner will withstand death, and if their grave is ever opened, the viewers in the future will be able to see the strength of their love.  Donne also provides a comment on the eternal nature of the written word, trusting that his poem will outlive them and future generations will know of their love from the poem itself: 

And since at such time miracles are sought,

I would have that age by this paper taught

What miracles we harmless lovers wrought.

As for the definition of the style, I found a reference from the Academy of American Poets (cited below) which defines three styles of odes: Pindaric, Horatian, and Irregular.  The Pindaric was longer and more complex.  Horace followed with a shorter, more straightforward style.  The Irregular is just the name for odes that cannot be categorized by the first two names.  "The Relic," in my opinion, is more in the Pindaric than Horatian style.  However, the lines between the categories are not clearly defined, so arguments could be made either way.