To get some figures around here:
Bounce Tour: 83 Shows / 72 times played - number of different songs played (incl. covers): 61
Have A Nice Day Tour: 88 / 47 - 73
Lost Highway Tour: 99 / 34 - 100
The Circle Tour: 83 / 5 - 95
Live 2011: 59 / 9 - 102
Because We Can Tour: 103 / 0 - 90

I think it's a mix of different reasons.
First of all the number of albums to pick the songs for the setlist raised each tour (of course).
With the Lost Highway tour the band started to vary the setlists much more than the tours before.
And I also think that it's a vocal issue as well. Just listen to the early versions and compare it to the latest versions. Jon struggles with the chorus and uses a different kind of emphasis to make it through the chorus.
On the studio and early live versions he sings the way it's supposed to be: "It's noooooot ooooold" with the long note on the "not" and a slightly longer note on the old.
On later version he sang "It's not ooold" with only a slightly longer note on the old so that he had more breathe for "just ooooooldeeeerrr". And later on even left out the "just" and only sang "oooldeeeer".
So I think that might be one of the main reasons to get rid of it more and more.
I don't think it has something to do with Richie. There'd be a whole lot of other songs he wouldn't play for that reason either (Blood On Blood, Whole Lot Of Leavin', I'll Be There For You [ok, at least a hit]...).