Readers should know at least two things about "Corona":
- it alludes to Rainer Maria Rilke's poem Herbsttag  and
- it is a love poem about Celan and Ingeborg Bachmann.
There are several echos and parallels between between Rilke's "Herbsttag" / "Autumn Day" and "Corona". The most obvious one is the words "es ist Zeit". However, Celan's "Corona" puts these words in the last stanza and leaves out "Herr" / "Lord". Two world wars took place between the composition of "Herbsttag" in 1902 and the publication of "Corona" in 1952 in the collection Mohn und Gedächtnis. (The collection took its title from a line in "Corona".) Of course, there is also Celan's personal history. Celan was Jewish and was away from home when Nazi troops came to his hometown Czernowitz in 1941 and deported his parents. His parents died in the following year and Celan never forgave himself for having "abandoned" his parents. (See for example another poem from *Mohn und Gedächtnis", "Aspen Tree", which contains the line "Meiner Mutter Haar ward nimmer weiß" / "My mother's hair ne'er turned white".)
In addition to the phrase "es ist Zeit", there are other parallels between the two poems: both are set in autumn and both want certain things to happen now rather than later. The autumn leaves that Rilke mentions in the last line of his poem appear in the first line of Celan's poem. In both poems, restlessness is mentioned at the end ("unruhig" in "Herbsttag", "Unrast" in "Corona").
Whereas Rilke's poem lists processes that the poet wishes to happen, especially in the first two stanzas, two first two stanzas in "Corona" appear to describe statis or the impossibility to make progress. Time is set free from its shell but returns to it; Sunday is a day of rest (at least in Western Europe); the dream is not about anything interesting or unusual but only brings more sleep. 
Ingeborg Bachmann met Celan for the first time in 1947. She also visited him in Paris in the autumn of 1950 and stayed in Celan's hotel room for a month, but living together didn't see to work. Both authors kept their relationship secret, and knowledge about it transpired only gradually after their deaths. Their correspondence was published for the first time in 2008 and shed new light on some of Celan's poems. (See Böttiger's review.)
The third stanza might be read as alluding to the chasm between the lovers. Celan was a Jew and Bachmaan a Christian. "Geschlecht" can be read as "(genealogical) lineage" (see Geschlecht on Wikipedia) or even "race" (in a biological sense, see Wikipedia discussion of the vague boundaries of the term "Rasse" in German). These meanings suggest a distance between the two lovers, whereas the translation "[my lover’s] sex" doesn't, or not in an obvious way. This distance might explain the gazing (which you can do at a distance) and the "dark things" they speak of.
The fourth stanza raises more questions. What does it mean to "love each other like poppy and memory"? We know from the poet's biography that he had painful memories as a consequence of World War II. If these are relevant here, the poppy, which can relieve pain, counteracts those memories. But from that point of view, poppy and memory don't "live together", let alone love each other, unless we interpret the lover as the pain killer that makes living with those memories bearable. This interpretation, however, seems unable to make sense of the tercet's two other lines, which read like surreal images induced by an opium dream.
In the fifth stanza, the lovers are together and perhaps it is time now to realise what the poet warned about in Rilke's "Herbsttag": build a house (perhaps metaphorically for building a family) so as not to stay alone, instead of restlessly wandering, as Celan had done to some extent after World War II, when he mover from the work-camp to Czernowitz/Cernăuți, then to Bucharest, then to Vienna and finally to Paris. So perhaps it is time that the "Unrast" (restlessness) get a "Herz", which may refer both to a regular heartbeat (contrasting with restlessness) and a loved one (as reminiscent of the phrase "[du bist] mein Herz"). This would allow time to progress again (contrasting with the statis in the first two stanzas) and bome a normal organising principle again. Hence, the penultimate line may be saying, "It is overdue that time can run its normal course." The last line then appears to collapse both these meanings of "time"; the difference doesn't matter any longer.
John Felstiner writes about stanzas four and five (page 54:
They ["we" from line 10] both must forget and remember their pasts, maybe because one of the is Christian and the other not. Also divided, the word "corona" suggests Christ's crown of thorns and a new time come round. The lovers' union has a civic dimension—"it's time people knew="—and a prophetic one as well: "It's time the stone consented to bloom." For Celan, who shoveled stones at forced labor and could not give his parents a gravestone, "stone" already stood for mute grief. If a stone is to bloom, there there must come—to borrow from "Late and Deep", written shortly before—a "wind blast of conversion."
Hugo Bekker has more to say about "it is time" (page 203; bold text from the original):
When, having trekked through Corona until its final "Es ist Zeit" and wanting to believe that it is indded time to make an important acknowledgement, we do not quite succeed. Already heralded by "es ist Zeit, daß du kommst und mich küssest!" of Die Hand voller Stunden (162), Corona's final strophe may relay a shrinking back from "es ist Zeit, daß man weiß", particularly if that impersonal "man" refers to the speaker himself rather than to the public in the street. (...) We therefore yield to the temptation to read the final "Es ist Zeit" as Est ist Zeit, daß es Zeit wird, daß es Zeit ist. The 'impossible' motif of the stone coming into bloom—representing the speaker himself—thus receives its motivation.
 According to a survey held in Germany in 2000, "Herbsttag" is the third best-loved poem among Germans. See Die Lieblingsgedichte der Deutschen, 11th edition, Artemis & Winkler, 2011. Rilke was well represented in this top 100; Celan also had a poem that reached the top 100, namely "Todesfuge" / "Death Fugue", at number 16.)
 The line "the mouth speaks true" doesn't fit this pattern, but this answer does not attempt to provide a full interpreation of the poem.
 The last stanza of Ingeborg Bachmann's poem "Das erstgeborene Land" / "The first-born land", published in 1956, may be read as alluding to and possibly a coded answer to the stone image in "Corona". See Huml, page 193.
- Bekker, Hugo: *Paul Celan: Studies in His Early Poetry". Rodopi, 2008.
- Böttiger, Helmut: "„Ich habe ihn mehr geliebt als mein Leben!“" [Review of Ingeborg Bachmann / Paul Celan: Herzzeit. Der Briefwechsel. Mit den Briefwechseln zwischen Paul Celan und Max Frisch sowie zwischen Ingeborg Bachmann und Gisèle Celan-Lestrange. Herausgegeben und kommentiert von Bertrand Badiou, Hans Höller, Andrea Stoll und Barbara Wiedemann. Suhrkamp Verlag, Frankfurt am Main. 399 Seiten.] Deutschlandfunk, 17.08.2008.
- Buch, Hans Christoph: "Als Corona noch ein Himmelszeichen war", Welt, 11.04.2020.
- Felstiner, John: Paul Celan: Poet, Survivor, Jew. Yale University Press, 2001.
- Huml, Ariane: Silben im Oleander, Wort im Akaziengrün. Zum literarischen Italienbild Ingeborg Bachmanns. Wallstein, 1999.