As a general rule, issues due to the total mass of the rider and equipment on the bike relating to tire pressure will be due to having the pressure too low, not too high. In such cases, you can end up with what is usually known either as a pinch flat or ‘snakebite’, resulting from pressure from the rims effectively puncturing the tire. This is why most bike tires usually have a minimum pressure rating in addition to a max.
In general, tire pressure should almost exclusively reflect the type of surfaces you are riding on, within the ratings for the tires (high pressures for solid and mostly smooth surfaces, low for loose or rough surfaces).
In terms of actual risk of damage due to overloading, the primary risk is increased stress on the spokes (especially on the back wheel, as that’s where most of the weight is on almost any bike), or possible long-term fatigue of joints on the (though the spokes will fail long before that becomes an issue). The biggest thing you can do to mitigate that (other than losing weight) is to be careful going over bumps. Stand out of the seat when doing so and use your legs as shock-absorbers (this will also, once you get used to it, probably be more comfortable than sitting in the saddle as well). Higher tire pressures can make this worse due to less cushioning, so you might consider running a bit below what you might otherwise run.
The other two major things to consider due to weight are increased break wear (more weight means more momentum, which in turn translates to more energy that you need to dump to bring the bike to a stop), and an increased risk of the wheels going out of true (due to increased horizontal loads while turning). Both are things that regular inspection and regular preventative maintenance can easily take care of though.
On the note of punctures, tire pressure usually does not matter much here. Lower pressures are less likely to result in less pointy objects embedding in the tire itself, but the practical impact of this is essentially nil.
If you are worried about the possibility of flats, there are a couple of options:
- Carry some spare tubes and the tools you need to replace a tube on the side of the road.
- Consider tire liners. Some bike accessory brands make plastic or fabric liners you can put inside your tires that will help protect against punctures. They’re generally decent at this and reasonably cheap, though they will add more weight to your wheels, can be finicky to install, and also marginally increase the risk of a pinch flat (because there is less space inside the tire for the tube).
- Look into more puncture resistant tires. Continental Contact Speed tires are supposed to be pretty well reinforced, but you can find better. Just keep in mind that something very sharp and pointy like a nail or tack can still punch through in many cases, it will just take it longer to do so.
- Look into ‘self-sealing’ tubes. These are a kind of halfway point between traditional tires and tubeless tires, consisting of an tube with tubeless sealant inside. The nice ones can be refilled just like a tubeless tire, the cheap ones can't but are much less expensive, either way you would need to replace tubes every couple of years.
- Consider something like Tannus Armour. Just like self-sealing tubes are a kind of halfway point between traditional tires and tubeless, this is a halfway point between traditional tires and airless tires. The feel of the ride on these is distinctly different from a normal tube, but they will protect you from almost anything, and if a tube does rupture you can safely ride on a flat provided you don’t go too fast.
- Consider just getting airless tires. These replace both the tube and the tire in most cases, and completely eliminate any risk of a flat. The downsides are that they wear out faster than regular tires or tubes, they are often a pain to install, and you lose the control of handling from tweaking tire pressure.