Prior to being upgraded with a Hard Light Bee, Arnold Rimmer's hologram was able to pass through solid objects such as walls, people, and desks. The Soft Light Light Bee inside of his hologram was proven to be a solid three-dimensional object when it was handled by both Dave Lister and Kryten.
How was Rimmer's hologram able to pass through solid things with a three-dimensional 'Light Bee' inside of it?
On several occasions we see Rimmer moving around the ship without the (evident) use of his light bee, strongly suggesting that the projection is remote rather than local.
Notably in the episode Thanks for the Memory he uses a 'Hologramatic Projection Cage' (see below) when his light bee is unavailable. It therefore seems perfectly plausible that this same technology was installed aboard Red Dwarf in certain areas of the ship.
This also occurs in the Series X episode Entangled where Rimmer briefly enters 'soft-light projection mode' in order to travel through a solid object. Both these events happen on board the Red Dwarf, suggesting that his hologram can (under certain circumstances) be generated locally or via a projection.
I would've thought in addition to the other answers that the Light Bee itself is possibly capable of remote projection, though only in soft light mode. In Demons and Angels when his battery starts going and he has to find somewhere to recharge we are reminded that away from Red Dwarf he is entirely dependent upon his Light Bee. One wonders, given that they were separated from Red Dwarf in series VI-VII what would happen to Rimmer if Starbug/Their link with Holly was destroyed? Lister has the Holly watch, but still if contact will the Holo simulation suite were totally severed it would seem difficult to account for Rimmer's continued existence.
I just figured that while using the light bee Rimmers entire program was stored in the light bee & was generating His entire self from the light bee, as seen in the S6Ep5 Rimmerworld where he's completely cut off from Red Dwarf & Holly via a wormhole. I just figured that in eps such as entangled where Rimmer goes into softlight mode & passes through objects with his light bee that the light bee was somehow able to go slightly out of phase with normal space & so could also pass through things while still projecting Rimmer.
It is likely the light bee was discovered between seasons 2 and 3 which is why Rimmer has far more freedom of movement from that point.
The fact that the M-Corps update to Red Dwarf not only didn't work on Rimmee but made him invisible to Lister would suggest that the bee is not part of Red Dwarf's own holoprojection equipment as that was all bought by M-Corps.
As for passing through objects later - according to sales info for the bee it can project a hologram through a solid object though it doesn't explain how this is achieved.