No, it seems to suggest just the opposite. The speaker can summon up remembrance of things past and relive them even long after he suffered from them emotionally, including "love's long-since cancell'd woe" and "precious friends" who have been dead for some time. The poem deals with the fact that Shakespeare, like ourselves, relives old events and has to suffer the pains they caused him in the distant past as if he had not already suffered them ("Which I new pay as if not paid before"). If an experience was important enough, and painful enough, we will all remember it for the rest of our lives. This is why Shakespeare's sonnet is so touching. In the concluding couplet he says that with thoughts of his "dear friend, / All losses are restor'd, and sorrows end." But this does not mean that the losses are premanently restored or that the sorrows are permanently ended, only for "the while."
"Remembrance of Things Past" was used as the title of the best English translation of Marcel Proust's A la recherche du temps perdu by C. K. Scott Moncrieff. In that great work the past remains accessible in its entirety and can actually be completely relived.