If the CPU temperature is unusually high on an older system or when removing/replacing the heat sink/CPU one should always replace the thermal paste.
But should one do so preemptively? If so, how often should one replace the thermal paste? And does the same apply to thermal adhesive pads/tape?
No. Unless you have thermal issues or instability that you can definitely link to heat, you do not need to replace thermal paste.
I have systems that never got the thermal paste replaced and still run fine after 7 years.
There are people who say that after 3 years one could renew the thermal paste to improve the performance a bit, and while this is probably true, it is possible, thermals were improved simply because better thermal paste was applied the second time.
I therefore still believe in: If it's not broken, don't fix it.
In general, no. You shouldn't do that unless you have issues with your cooling system. Most (except the cheapest) thermal pastes are very durable and barely dries up when secured between the CPU (or GPU) and the heatsink, even after 10 years or so.
I personally have used products like Arctic Silver 5 and Noctua NT-H1, both of which have excellent durability, though the former is too thick to apply by normal means.
Also note that most of the time when you reapply the paste you'd also clean the dust from the fans, which typically improves the cooling performance much more than reapplying the paste itself. This is likely where most such claims come from.
But should one do so preemptively? If so, how often should one replace the thermal paste?
Normally the answer is no and to wait till there are signs of thermal throttling or stability issues like @LPChip said in their answer. If your paste is old, and you know that you're going to put a lot of thermal cycles on it, (like AI image/video generation that shifts between the CPU and GPU) for fast short periods, then you could consider it. If something like that isn't going to generate you money the faster it's done, do you really need to replace the paste? If you have the money, time, and/or the desire to do it, that's up to you. Most CPUs/GPUs will have mechanisms to detect when they're getting too hot and throttle themselves.
And does the same apply to thermal adhesive pads/tape?
This one tends to be a no, even if you remove/replace the cooler. There are pads such as Thermal Grizzly's KryoSheet, (no affiliation), which I have used for my current system. I had to remove the cooler a few times troubleshooting RAM issues, and the pad still removes heat like a champ. This type of pad is basically a cloth with graphene in it to pull heat away and be reused. Drying out is not possible because it's already dry. It can't seep out because it's basically a cloth.
Then there are products like the: Honeywell PTM7950 Phase Change Thermal Pads, (no affiliation). This is also a no and a bit different as it will change from a solid to a liquid depending on how hot it is. There's no water in it to dry out, but it can seep out from the cooler a little if the pad is too big when installed. After that it tends to stay place from what I can tell. I've read that you can reuse it if you don't remove it from the CPU or cooler, there might be some air in the pad but that should be expelled after a few heat cycles, hopefully.
For thermal tape, I would not use any of those. They tend to have a very low thermal conductivity listed on their spec sheet.
There are three reasons that you need to replace your thermal paste: dry out, pump out, breakdown. The paste you choose will determine if you can get these issues, and how long it will take before signs of the issues show up.
For dry out, most pastes have some kind of liquid in them to keep the solids bound together and able to flow easily. After a while it might evaporate and the paste will become hard and shrink a little, leaving gaps that aren't as effective.
Pump out is when the thermal cycling causes the paste to be pushed out from between the the device and cooler. Since the paste was pushed out, there's nothing in between the cooler and the device.
Breakdown is when the thermal compound's chemical compounds permanently change and is no longer as effective as you'd like it to be. Time and lots of thermal cycles can do this, highly depends on your paste though.
Pads mostly avoid these issues, but tend to be a lot more expensive.
Bonus: Liquid Metal
Now here's a mixed bag. Liquid metal has really good thermal conductivity properties; however you have to make sure that the device and cooler you're using are compatible with the liquid metal that you're going to use. If your heat sink's base is aluminum, you need to make sure that the liquid metal is safe to use with aluminum, or the heat sink's base or more will be broken down by the liquid metal.
Edit: When using liquid metal, keep in mind that it can seep/leak out very easily from between the device and cooler, and can conduct electricity. If it leeks out onto any unprotected components, it will short them out and probably destroy them or cause other problems. For example: Sony's PS5 uses liquid metal for the CPU, and it has a gasket around the CPU die to prevent leakage, (no affiliation to iFixit).
