Is it safe to have it turned on 24/7?
Yes.
Should I remove the battery from the laptop or keep it as a backup power source?
I'd keep it. The laptop will run off power supply if it's connected. The excessive battery wear when laptop is running on AC is mostly a myth. With default settings the battery will charge to 100% and then laptop's circuitry will disable charging until charge level drops below 90% to conserve battery life. So you can safely keep the battery inserted and use it as a backup power source.
Lithium-polymer batteries should in general be stored at 60% charge if not used, but custom charge threshold setting is rarely present in consumer grade laptops. So you can either store the unused battery in perfect conditions or make use of it. The latter makes much more sense in my opinion. Keeping the battery inserted and charged to 90-100% won't be dangerous to the computer, it will just shorten battery life compared to storage at 60%. (Thanks to @davidgo for reminding me about this)
Also, is it safe to run HDD 24/7 or should I enable APM so it spins down after some time of inactivity?
It's not just safe, but recommended! HDDs are more likely to fail when spinning up than when running continuously. Spinning disks down is useful to conserve energy or reduce noise, but it will affect disks' longevity. For a server it's better to keep them spinning constantly.
Can the laptop burn if I leave it running?
Not if the cooling system is working properly. Even if it's not, thermal throttling will kick in and reduce performance to keep temperatures in safe range. A properly built computer should run 24/7 no problem.