You also run ultra-cold (thanks for that tip in the other thread btw).
The question wasnt really whether they raise heat or cause problems for some (they do, even if not much for certain setups), it's WHY they do.
So I finally went to a neutron star today (HIP 47126), and I came to a few revelations that make me think the game is actually being rather consistent and scientific in the case of neutron stars. It isn't the radiation that is causing heat damage by itself. The radiation is simply preventing the ship from cooling off while using any heat generating ship function. So really, the ship is damaging itself from inefficient heat venting when in proximity to a high energy density field. This is why damage is only instigated by dropping warp, boosting, or spooling FSDs. It isn't the motion through an EM field that kills you, it's the nuclear engine overheating from within.
I had also originally thought that the game was acting as if Neutron Stars were more dangerous from a great (or similar) distance than a normal star. However, the FSD-drop distance from a neutron star is ~100 times closer than that of a normal star. The Neutron Star only seems far away because it is so incredibly small (eg the size of a
City in modern day earth). If we were this close to a normal star we would be getting 10,000 times more radiation energy than we do at the current FSD-drop distance.
For illustration, this how close we drop a normal star:

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And this is how close we drop to a Neutron Star:

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Yet the heat energy from a neutron star is only slightly higher than that of the average star at this extremely close range. So if radiation field around a Neutron was 2 times hotter than the average star, that would mean that the NS was only emitting 1/5000th the energy output at an equivalent orbital radius. And this is actually the average output that we see in nature from naked neutron stars. This should demonstrate how dim the Neutron star is despite it's much higher surface temps. So, top points to David Braben and Co. They really did their homework on this one! I am continually impressed by the level of detail and realism they put into this game.
As for the heat damage from a black hole? If there is no accretion disk and no radiation coming from it, then I can only assume that heat damage is a proxy (or place holder) for tidal stresses acting on the ship. Either that or the potentially strong magnetic fields coming from a black hole might disrupt the power grid of the ship causing it to work less efficiently and generate more heat per MWatt, thus causing it to run hot, and easily overheat.