If the coffee is just a day or two old you should not get sick from it.
But if the coffee is old enough to have mold growing in it then yes you could get sick from drinking old coffee that has grown mold inside it.
I sometimes have a pot of coffee left over in the evening before going to bed and then just reheat it in the morning by pouring a cup of coffee in a ceramic coffee mug and then heating the coffee back up in the microwave.
Although the coffee isn't as fresh when reheated after sitting overnight the coffee is still good and safe to drink.
I also like to put a little coffee into some ice cream which makes it taste good too.