Thanks to B-R.com's Play Index, we can see that, since 1957, there have only been six major league games played with a -handed second baseman. Mattingly shows up, even though he only played one half-inning. There's also Gonzalo Marquez, who played two games at 2B for the Athletics in 1973. Here's the first game, in which Marquez reached on an error in the top of the first and then A's manager Dick Williams replaced him for the bottom of the first inning, so Marquez never saw the field. The next day, Williams did the same thing, using four different players at 2B in the game. So Marquez never fielded a ball at second either.
The next entry is Thad Bosley, an OF/DH for the Royals (among other teams) in the 1980s. A -handed thrower, Bosley pinch-hit for the starting second baseman (Buddy Biancalana) in the bottom of the eighth inning. He then stayed on and played the top of the ninth and did not see a play at second base, and was later replaced for a pinch hitter in the bottom of the ninth.
The oldest case listed is George Crowe, a first-baseman with the Cincinnati Reds who, in a 1958 game, was briefly switched with second-baseman Johnny Temple for one play. Following a single by Bobby Thomson (yes, of shot-heard-round-the-world fame), the two switched positions for a play cryptically described as a "Double Play: Pop-fly, 1B to 2B." What happened here? Was there some kind of trickery going on? The two men switched back after the play and Crowe never played second base for the rest of his career. Since there are no reliable box scores prior to 1957, we do not know if this sort of thing happened a lot.
Finally, the strangest case of all, that of Sam McDowell, -handed pitcher. On July 6, 1970, McDowell was pitching for the Indians in the eighth inning of a game agains the Senators with a 6-4 lead. To that point McDowell had scattered 11 hits in 7.2 innings striking out 12 while allowing the four runs. With runners on second and third and two out, Cleveland manager Alvin Dark called to the bullpen for a righty, Dean Chance. Powerful righty Frank Howard was up next, followed by righty cleanup hitter Rick Reichardt. But Dark didn't send McDowell off the field; instead, he sent off third baseman Greg Nettles and told McDowell to go play second base. Chance then intentionally walked Howard to load the bases, then induced a groundball to third from Reichardt. Eddie Lemon, who had been playing second base, fielded the ball and threw to McDowell at 2nd for the force to end the inning. McDowell would then re-take the mound for the ninth, striking out the side to end the game. I'm not sure it's officially in the rules somewhere, but I suppose one cannot be charged with both a Win and a Save in the same game...