Short Answer: No.
In the chapter 9 of the player's handbook, there is a section titled "Damage Rolls", which covers the general rules for dealing damage with weapons or spells.
Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any m odifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage.
When attacking with a weapon, you add your ability modifier—the same modifier used for the attack roll—to the damage. A spell tells you which dice to roll for damage and whether to add any modifiers.
If a spell or other effect deals damage to more than one target at the same time, roll the damage once for all of them. For example, when a wizard casts fireball or a cleric casts flame strike, the spell’s damage is rolled once for all creatures caught in the blast.
According to this rule, the spell description should tell you if the spell's damage requires adding an ability modifier. Different spells do different damage, and not all of them add a modifier to the attack.
In most cases, however, spells do not require the addition of your spellcasting ability to their damage. Notable exceptions are Eldritch Blasts cast by a Warlock with the proper invocation, any evocation spells cast by an Evoker of level 10 and above, some elemental spells cast by a dragon-blooded sorcerer, or some specific spells cast by clerics of some domains.