Yes, heat can damage your laptop. HDDs start to wear out at 50C, and a lil higher and it will create its own vibration that'll begin to damage the spindle. CPUs are usually rated to around 90C but set the limits at 70-80C (before throttling or emergency shutdown). GPUs are probably rated at around 100C (they're expected to run hotter) and 60-80C is pretty normal for high end laptop graphics. Enthusiasts aren't happy if the GPU runs over 65C though, so they'll take a number of measures to keep temperatures down.
First and most important bit of advice: maintainance. Clean out the dust. Do this every month. Dust is one of the 3 worst enemies of laptops (along with heat and movement), and it will build up no matter what you do (clogging the vents, lining the heatsinks, weighing/slowing down the fan, and potentially creating static fields). Clean it out, and be thorough.
Enthusiasts tend to take it a step further and replace the thermal paste on the CPU/GPU, although this is only recommended if you've enough experience in taking apart computers to know how to put it back together again without breaking anything. Cleaning the dust and replacing the thermal paste can drop temperatures by around 6-15C (in extreme cases).
After that, as mentioned, get a cooler. A cooling pad to sit the laptop on has become something of an essential buy for stardard/high-end laptops, and some go a step further to buy USB fans to point at the top of the laptop (keeping the keyboard cool). The point of this is that laptops have so little space inside that extra coolers from outside can help the working environment. Stating the obvious, but still.
Otherwise there are some software/OS tweaks you can make to reduce demand on resources, as well as underclocking/volting the GPU to make it run cooler all the time. These things tend to take time and patience to seek out and set up to the right configurations.
