From what I know, LBS does keep a track of the applicants who have visited the school. However, the admissions committee is well aware that it is not possible for everyone to visit the school and hence it is no way a deciding factor when it comes to making a decision. The decision is solely based upon your application and that of other applicants.
I think it would be worthwhile to travel to NY or DC to attend an information session since it would give you first-hand information about the school and its curriculum. Plus you get to meet alums and admissions staff and in case you have any queries, they can be resolved right there and then.
Let me know if you have any other questions regarding the programme or the school.
AppMentor - accepted students' advice