Statistically there is enough proof to say yes, but some teams do not let it get to them. The thing with playing at home is you are feeding off your home crowd, you are able to get into a set routine, and you are able to get a great feel for your position in the field.
Of course home field advantage matters in baseball. The sport is a numbers game and if you were to look at the numbers you would see that home teams win more than road teams. Also, players are actually playing harder now in the All-Star game because it determines who has home field during the World Series.
Anyone who doubts this has not looked at win percentages over the years at home versus on the road. On a consistent basis, teams do better at home. They have fan support, they feel comfortable, and they know the quirks of the ballpark's field and are even adjusted to the weather. There may be individual players and teams who on occasion do poorly at home, but the overwhelming evidence shows that it is an advantage to be the home team.
It is there, and it does exist, but not to the extents of other sports. The relationship between fans and the on field play is much farther removed than from other sports. Baseball is a much more routine sport, and fans in baseball are much quieter overall, and pressure doesn't get to baseball players as easily.