"And Death Shall Have No Dominion" is about resurrection. The body dies but the human spirit lives on. The spirit lives on in nature itself. Thomas doesn't delve into abstract notions of immortality or the soul. His religious framework in this poem is pantheism, a view that spirit (or God) and nature are one. One need not look to abstracts for the spirit because spirit and nature are all ("pan") one. Thus, in the first stanza, the dead "shall have stars at elbow and foot." The spirit survives the death of the body and remains in nature. And it doesn't matter where in nature; in this sense, spirit is anywhere/everywhere. In the same stanza, "lovers be lost love shall not" indicates the same sentiment: lovers (people) may be lost but love itself is not.
In the second stanza, the speaker notes that neither torture nor a loss of faith shall kill the spirit. Thomas stays with the pantheistic theme. One doesn't have to have faith in a particular religion. The spirit will live on regardless of belief. It is possible that this is a rejection of Christianity or organized religion in general.
The last stanza suggests that the immortal spirit can be revealed in other things in nature, such as the daisies or even the sun. Further, even when these signs of life (and spirit) die or end, the spirit will live on in other manifestations. The earthly body may be broken, but the spirit remains whole as it is one with nature. Even mental breakdowns will not diminish the spirit. "Thou they go mad they shall be sane" in the first stanza and broken faith in the second stanza suggest that even these mental breaks do not affect the spirit. This is important to note because faith is sometimes conflated with spirit. The speaker in this poem does not conflate the two. In this poem, the speaker is probably describing faith as a particular belief system and spirit is not framed within any belief system. Spirit is a force which is somehow abstract but residing in nature.
The title, and refrain, refer to a Biblical passage, Romans 6:9, "For we know that since Christ was raised from the dead, he cannot die again; death no longer has mastery over him." You will find variations of this passage. Some have the exact words "and death shall have no dominion." Although Thomas may be rejecting Christianity in this poem, it is likely that he got the title from the Biblical passage. So, it is unclear if it was a rejection or if, in this pantheistic image, he used an element of Christianity that suited his purposes.
Also note that the speaker is not someone in particular, as if it is a disembodied voice coming out of the air. This underscores the idea that the spirit (voice) is one with nature (the air).