Usually no. The animals most often called daddy longlegs in North America are harvestmen (order Opiliones)—arachnids with eight long legs and a compact, apparently single-segment body. They are not true spiders (order Araneae): spiders have two distinct body sections and typically produce silk, while harvestmen do not build webs and are more closely related to scorpions than to spiders.
Confusion arises because the same nickname is also used for cellar spiders (family Pholcidae), which are true spiders, and sometimes for crane flies. Neither true daddy longlegs nor cellar spiders are dangerous to people.