If I leave my computer on all day without turning it off/rebooting it for months, is there a possibility that it might break or something?
I hate turning it off, but I don't want it to break.
I suppose the mechanical parts (fan, disk drive) might wear out sooner, but their design life is so long, a small reduction isn't going to matter. As for the electronic parts: assuming that they're adequately cooled, usage is not going to wear them out.
Still, you are wasting energy. If you don't care for the inconvenience of a cold boot, consider using sleep or hibernate mode which preserves your system state while the machine is turned off.