Basically, it boiled down to two things:
In one case, a particularly strict anti-tipping law was found to be discriminatory and therefore unconstitutional. (Such a law might be called "classist" today.)
The laws didn't change people's behavior anyway.
With a hat tip to @user6726, the following is drawn from Tipping: An American Social History of Gratuities, by Kerry Segrave. From pp. 37–38:
Iowa's law was particularly onerous in that sanctions were applied only to tip takers who were employees, while a business owner was free to accept gratuities. ... In Iowa, a barber by the name of Dunahoo was arrested, charges, and convicted after a customer gave him a 25 cent tip after a haircut. The shop contained two chairs, one staffed by Dunahoo, the other by shop owner Murphy. Had the customer chosen Murphy and tipped him it would have been legal since Murphy was a proprietor, not an employee. Dunahoo's arrangement with Murphy was that he was to receive a salary of $15 a week plus 60 percent of his billings above that amount, plus tips. When the case went to the Iowa Supreme Court, it turned on the discriminatory nature of the bill, which, contrary to the United States and Iowa constitutions, denied equal protection under its laws to all citizen — separate classes were established. Dunahoo's conviction was reversed by a 4–2 margin. The majority decision said, in part, "Tipping may be an evil, but this does not justify discrimination between classes in order to put it down. In so far as the public is concerned, the evil of tipping the employer is quite as obnoxious to good morals as though it were done to the employee."
By the end of the 1910s, the fever to pass antitipping laws had waned considerably. Washington's law was repealed in 1913, largely because it was ignored by all sides. ... The remaining state acts were repealed as follows: South Carolina in 1922, Tennessee and Arkansas in 1925, and Mississippi in 1926. Although antitipping sentiment was still strong in the 1920s, those acts were repealed partly due to the fall of the Iowa measure, and partly due to the persistence of the custom, which led many to believe it was futile to try to abolish the practice.