TL;DR Your system will be fine if you leave it on all the time.
Okay, I'm a system administrator and used to do field IT work. Some of the other answers seem to be a bit misinformed. I personally leave my work laptop on 24/7 and it's not a server. I also have a custom gaming rig at home that I leave on all day as well, it's current up time is 45 days. While systems will build up dust on the fans the longer you leave them on, the CPU and certain other parts are designed to slow themselves down to avoid overheating. Pets, smoking, dusty roads, and the like can increase the likely hood of dust build up. If you notice a drop in performance, a good cleaning, (OUTSIDE of your house), might help it out. Use a can of compressed air and avoid touching the electronic parts inside of the computer.
As for being on the net 24/7, that all depends. If you're on a website that refreshes itself and the ads that are displayed, then it is possible to get an infection. Closing your browsers can keep this from happening. Most of the malware I've had to remove from systems come from from the ads on websites. Using either Firefox or Chrome with Java disabled and Flashblock can reduce your chances of getting malware greatly. Internet Explorer and Edge are both jokes and I recommend you avoid them like the plague they are. One final note about being on the internet, if you are not using a router, and you're directly on the internet, then you are at risk. If your modem from your ISP only has one network cable port coming out of it and it is going directly into your computer, that probably means that you're not behind router and you're most likely exposed. While yes your computer probably does have a firewall, it's probably not going to be as good as a router's, (especially if you're only using the Windows Firewall). If you can plug your network cable into one of many ports on your modem, then you're probably fine.
As for the battery in your laptop, do NOT remove the power when the battery is fully charged. Your laptop is most likely smart enough to know when the battery is charged and it will stop charging the battery once it's full. The battery is like your phone's battery, the more you completely drain it, the faster it will wear out. If you have an external battery, such as a UPS (Uninterruptible Power Supply), then you can remove the laptops battery altogether, which will help with the heat.
When looking at the answers about wear and tear, most of the components in PCs are like light bulbs. Turning them on and off regularly will wear them out faster than constantly leaving them on. A good number of the parts in your PC will get to at least 140°F/60°C during normal use, (CPU, GPU, certain bridge components, etc.). This is why cooling is so important. After shutting down the system they will naturally cool back down, so it's like bending a paper clip back and forth until it breaks. Capacitors are one of the bigger exceptions to this. They can wear out just by sitting there since they usually have a liquid inside of them that can leak or evaporate out.