What's he saying?
"No longer mourn for me when I am dead / Than you shall hear the surly sullen bell"
Stop mourning for me immediately after my funeral
"Give warning to the world that I am fled / From this vile world with vilest worms to dwell:"
Ring the death bell to let everyone know that I have died:
"Nay, if you read this line, remember not / The hand that writ it, for I love you so,"
If you read this poem, don't remember me, because I love you so much,
"That I in your sweet thoughts would be forgot, / If thinking on me then should make you woe."
That I don't want you to remember me if it causes you pain.
"O! if, I say, you look upon this verse, / When I perhaps compounded am with clay,"
But if you do read this poem when I am dead,
"Do not so much as my poor name rehearse; / But let your love even with my life decay;"
Don't even repeat my name; but let your love die with me;
"Lest the wise world should look into your moan, / And mock you with me after I am gone."
Otherwise, people will figure out why you are mourning, and make fun of your for being my friend.
Why is he saying it?
This and the following three sonnets deal with the poet's death; the speaker wonders how his memory will affect the fair lord after he is gone. The very existence of Sonnet 71 presents a paradox, since it is asking the fair lord not to remember his poet friend, but in order to know about this request, he must read the poem. Thus, in reading the poem, he will be remembering the poet. The poem can also be interpreted as a kind of role-reversal: the fair lord so often abandoned the speaker while they were both alive, and the speaker abandons the fair lord by dying.
The reason given in the final couplet for the fair lord to forget the poet and not mourn his death appears rather weak: the criticism of others is hardly a reason not to mourn a friend. However, Sonnets 57 and 58, which discuss the speaker's plight while he waits for attention from the fair lord, who prefers to spend time with other people, suggest that the opinion of the world is, in fact, very important to the fair lord. Thus the final couplet of Sonnet 71 can be seen as a bit sarcastic, pointing out the fair lord's shallow nature.
Sonnet 71's characterization of the world is contradictory: in line 4, it is referred to as "this vile world," but in line 13 it is called "the wise world." The whole tone of the poem suggests that the latter description is ironic, since the world will be prying into the fair lord's mourning in a nosy, annoying way. The idea that the world is vile is supported by Sonnet 66, which outlines all the vices of the world and the ways it has disappointed the poet, finally concluding with, "Tired with all these, from these would I be gone, / Save that, to die, I leave my love alone."
The speaker's memory seems to fade away bit by bit throughout the poem. In line one, he refers to himself as "me," but by line 6 he has become merely "the hand that writ" the poem. Line 9 reduces the memory still further to only, "this verse," and line 10 looks ahead to "When I perhaps compounded am with clay." This refers to the poet's body being reduced to dust, mixing with the clay in which it is buried. Finally, in line 11, he is only a "poor name," which he wishes the fair lord would not even speak aloud.
The state of the fair lord's memory of the speaker is likened to the speaker's decomposing body. In line 4, the poet describes how his dead body will decompose with "vilest worms;" the sentiment of other sonnets suggest that this is how the fair lord views the poet even in life. In line 10, the poet urges the fair lord not to think of him "When I perhaps compounded am with clay," or combining with the dirt in which he was buried. It is as if by this time, the fair lord's memory of him has become covered in figurative dust. Then in line 12, the poet makes a direct plea that the fair lord "let your love even with my life decay."