What I would like to know is how does an entity (stream of consciousness)...
I have never the Buddha taught about a stream of consciousness, let alone a stream of consciousness is an 'entity'. My impression is the doctrine/dogma/superstition of a stream of consciousness belongs to the commentaries & sects of what became later known as "Buddhism".
experience suffering if it doesn’t have a physical body?
I never read The Buddha teach about the above. For example, there are about two suttas from thousands (eg SN 4.23) where the Buddha says Mara (Satan) is looking for the consciousness of a dead monk. However, these suttas appear to not say if it was the Buddha who believed in a "consciousness after death" or Mara. However, based on the literal reading of these suttas, the only evidence we have is it is Mara (Satan) that believes in a disembodied consciousness. For example, there are many suttas, such as SN 38, SN 22.53 , SN 22.67 & SN 22.56.82 that say the Buddha said consciousness cannot exist without a physical body.
Like say in the hell realm or any of the lower formless realms. Is there anything in Buddhist teachings that can explain this phenomena?
Yes. SN 35.135 says "hell" is painful undesirable feelings felt at the six bases of sense contact.
I am also aware that mental suffering can intensify physical suffering, but I don’t understand how mental suffering alone, without a physical body is experienced in the formless realms.
The 1st noble truth says all suffering is attachment to the five aggregates. Attachment is clinging or possessiveness. For example, if I insult you, you suffer because your pride is wounded. This suffering is not related to pain of the physical body but it is the experience of mental pain (called "domanassa"). The pain of hurt pride is "formless" suffering.
Similarly, if the mind enters into the formless spheres of meditation (of infinite space, infinite consciousness, nothingness, etc), even though the physical body cannot be felt, if there is attachment to those formless spheres, there will be suffering. Any attachment, grasping or possessiveness is immediately stabbing or suffering; a loss of freedom.
SN 38.14 says there are three types of suffering:
(i) Suffering due to/about pain
(ii) Suffering due to/about change
(iii) Suffering due to mental proliferation, neurotic-psychobabble or monkey-mind.
Only one of the above three types of suffering is related to physical pain.