Nor Mars his sword nor war's quick fire shall burnThe living record of your memory (7-8)
This poem is what the speaker calls a "living record" of his beloved's memory. It's living for two reasons: (1) because it lasts forever (while other things crash and burn) and (2) because whenever someone reads this poem, the beloved will live within the reader's memory.
[…] your praise shall still find roomEven in the eyes of all posterity (10-11)
Punny Shakespeare. In addition to being a pun on "stanza," which means "room" in Italian, room also refers to the eyes of all the readers who sit down with Sonnet 55 and a cup of coffee. They will soak up the praise in these lines like a croissant in espresso and remember this beloved man with love and admiration.
[…] all posterityThat wear this world out to the ending doom (11-12)
Here "posterity" again means anyone who reads Shakespeare's sonnets—and given their popularity over the past 500 years, that's actually a lot of folks. Looking into the future, the speaker imagines how, after enough generations, all these people will eventually see the world to its end and bring about the final judgment. This may sound like a bummer, but remember that this poem transcends any ole individual death. As long as some readers are alive to read this poem and remember, the beloved will never truly die.