Was there ever a time or place when royalty, members of the ruling class, or wealthy people were legitimately transported in chairs, hoisted on the shoulders of slaves, like Xerxes in 300, Game of Thrones, or in old-time Hollywood films?

Yes, you're describing a litter chair (aka sedan chair, or palanquin) a form of human transport used throughout the world and over several time periods.

I am no expert, but while you are waiting for an answer these threads may give a bit more information and highlight the variety of cultures which used this transportation:

https://reddit.com/r/AskHistorians/comments/4o00r6/multiple_slaves_carrying_their_masters/ (answer by /u/qsertorius )

https://www.reddit.com/r/AskHistorians/comments/17blcv/ive_heard_in_medieval_japan_wheeled_transport_was/ (answered by /u/AsiaExpert )

https://www.reddit.com/r/AskHistorians/comments/2y7205/if_i_lived_in_rome_during_the_reign_of_julius/ (answered by /u/XenophonTheAthenian )

EDITED to ping users who answered original/relevant questions