I read somewhere that for example the neutralisation reaction between sodium hydroxide and acetic acid is less exothermic than those of sodium hydroxide with hydrochloric and nitric acid because some energy is needed to cause the weak acid (acetic acid) to completely ionise. Is this true?
I think the answer is closer to what was stated in the original question: "some energy is needed to cause the weak acid (acetic acid) to completely ionise". Stated differently, less energy is released from making acetate ion from acetic acid than from making chloride ion from hydrochloric acid (or water from hydronium ion).
Despite the lesser tendency of acetic acid to ionize, the overall stoichiometry of the two reactions is the same--as pointed out in a comment by the OP.