Does a harmful touch spell require an attack vs AC in order to succeed? If it does, what kind of roll should the caster do?
This depends on the spell. Read the description carefully.
Every spell is different and each spell contains instructions for how to use it. Touch spells are not any different in this regard. Touch only indicates the range, the target has to be within your natural reach (5' for a medium or small character).
If a spell requires an attack it will say "Make a melee spell attack", if it requires a save it will say "X effect happens" or "You deal X amount of damage" and then "The target makes a #type save" where #type can be any of the 6 stats.
Contagion (PHB page 227) - Make a melee spell attack against a creature within your reach
Bestow Curse (PHB page 218) Target must succeed on Wisdom saving throw or become cursed for the duration of this spell