While this point is debated among scholars, a strong case can be made that Petruchio comes to love Kate.
Admittedly, Petruchio comes to Padua, in his own words, "to wive it wealthily," (act 1, scene 2), meaning he is planning to marry for money. He goes on to say that for him, a wealthy marriage will be a happy marriage. He seeks out the shrewish Kate when he sees that all the other suitors flock to her docile and typically feminine sister.
Petruchio is the type of person who invests in relationships. If he simply wanted Kate for her money, he could have married her, gotten control of her money, and left her to lead a separate life, having a marriage in name only. This would not have been an unusual strategy in a time before companionate marriages were as highly valued as today. However, he instead invests time and energy in trying, using humorous reversals and exaggerations, as well as more stringent measures, to form her into a "fitting" partner. This shows he sees worth in her and that he perceives beneath her aggressive facade a person who wants to be loved.
Petruchio's trust in her at the end of the play also attests to his having positive feelings for her.
All of this may not add up to head over heels love by the end of the play, but it shows that Petruchio is heading down the path toward love.
There is another reason, too, to believe that Petruchio is in love with Kate. The play we see, in which he tames the shrew, is actually a play within a play. This points to the fictive nature of what happens: this is not meant to be the way things work out in real life but the way they do when we can write a fiction that matches our fantasy. Women, as Shakespeare knew, are not easily made docile—he is presenting a male fantasy in showing Kate tamed. Ideally, part of the fantasy would include the man falling in love with the woman he married and made "civil."