TOSHIBA Satellite CTX EZBook 300 Series, 400 Series, 500 Series, Series 700, 700C Series, 700M Series Laptop AC adapter
The common problem that's noted here is of overheating. Any electronic device or component heats up when electricity is constantly passed through it and laptop AC chargers and the laptops themselves are no exception. There have been cases where laptop AC chargers have burned out when used for too long. However, this is not the rule-there are several people who use their laptops over their AC adapters constantly and still nothing untoward happens.
90W HP Compaq NC4000 Laptop AC Adapter
The bigger fear is that the laptop itself might burn out. Now, while that's a possibility too, it is one that can be easily averted by providing good ventilation for the laptop. This can be done by adding a laptop fan which will help to dissipate the heat.
19V 4.22A 80W Fujitsu Lifebook S221O Ac Laptop Adapter
Another problem that might happen is the burnout of the battery. Though the AC adapter mightn't be harmed directly, it might direct an excessive amount of electricity into the battery which might retard its functioning. Laptop batteries work in terms of charging cycles and such constant electricity supply can be harmful to their longevity. Moreover, the laptop batteries might get used to the AC adapter and might not function as effectively as they did when they were newly out of the shop. For instance, if your battery gave you a charging of 3 hours earlier, consistent AC adapter use might reduce it to 2.5 hours. It is therefore advisable to remove the battery from your laptop if you intend to keep it on for a long time directly through the AC mains.
There are other things to consider, such as the excessive wear and tear of the components. Laptops, after all, are built for portability and that implies they will be rebooted often. That's the reason most laptops won't be able to take the pressure of constant use through the electricity mains directly.