An unmodified machine includes a large metal RF shield which usually performs the duty of being a heat sink for VIC-II, SID, and other chips.
I've worked on dozens of C64s, and have only seen the metal heat spreader/RF shield used in later C64 models, for example the 64C. Earlier "breadbin" models had a foil-covered cardboard RF shield that effectively held in heat, rather than to distribute it. However, these models did have the VIC-II inside a metal box with a heat spreader.
If you have a model with the metal heat spreader, that should provide sufficient heatsinking of the chips which tend to run warm. I'd suggest replacing the thermal paste, as it's probably dried out after all this time.
What is a reliable modern replacement for the original RF shield's heat dissipation function?
I find that 14x14x6mm aluminum heatsinks work well, attached using thermal double sided adhesive tape. I typically put 2-3 of these on the larger chips, depending on how hot they run.
Which chips in the C64 and C128 should have such a heat sink installed?
You can determine this by removing the case and letting the computer warm up for 10-20 minutes. Carefully check the temperature of each chip using your fingertip. If it feels hot, put a heatsink on it.
On an earlier model C64, I'd suggest heatsinking the following chips: CPU, SID, PLA, VIC-II (if it doesn't have the metal box/heat spreader), the 3 ROMs, and both CIA chips.
I'm not as familiar with the later "short board" 64C or 128, so I'll let someone else answer that part.