Yes and no. I know that's a strange answer but bear with me!
In terms of the show they try and keep it as close as possible. Just to give two examples from the soundtrack, Iago's big note in Diamond in the Rough was cut around the time the show opened in the UK and was cut from Broadway too in order to make the shows the same. The staging (and lyrics) of "though he's sitting here, watch Al disappear" in Friend Like Me was changed for the US tour and those changes were then implemented in the other productions as well.
That said, the humour and references (especially for Genie) do change for each production and even from Genie to Genie in the same production as the actors are encouraged to make the role their own and find what works for them.