There are lots of conflicts to be found in "Henry Manley, Living Alone, Keeps Time." Here's the one that stands out most clearly to me: the conflict between physical body and intangible spirit or soul. Henry Manley's body is aging and he is losing things physically—he's lost weight ("Knowing how much he weighed once / he knows how much he has departed his life."), and he is "as loose in his skin as a puppy." Being old and forgetful, things fall out of his pockets sometimes without him noticing.
The conflict comes when he loses intangible things by physical processes. The words he knows and loves are not physical objects, but ones that belong to his mind (his spirit?) and they too are disappearing as if they were tangible: "Window, wristwatch, cup, knife / are small prunes that drop from his pockets," and "the names of his dear ones / fall out of his eyeglass case."
This losing of intangible objects by physical processes foreshadows the loss he fears most: the loss of his non-physical soul from his physical body, no matter what holds him in (skin, blankets, etc.).
Especially he knows how the soul
can slip out of the body unannounced
like that helium-filled balloon
he opened his fingers on, years back.
The conflict between the physical and intangible appears elsewhere in the poem, too. The physical space is described vividly, but almost all the objects have an intangible significance beyond the physical. For example, the portrait of Henry's father is just "an oval on the wall," but it also represents a loved one whose name he forgets, despite the fact that he is staring directly at his father's face. Thus the spirit/soul/intangible slips away over time, no matter how much we work to tie it to us physically.