Well, firstly I'd reckon they'd be very reluctant to do it, unless they could seriously prevent people hacking in "designer pokémon". There are also issues of the target audience (still young kids basically) having access to MMORGS.
However, I don't reckon they'd stray from the old formula too much - wander around different areas, catch wild pokémon and battle trainers (many of whom would be other players). Obviously they would be able to run Leagues without any AI players, and quite possibly Gym Leader could be a player ambition (the Anime implies that there are many more than the 8 standard Kanto gyms - Gary has badges from other ones...). Grand Champion of the whole World? Yes please
Online would give Nintendo the ability to release legends and other rare pokémon without having to pre-determine the rules or trigger. They could set up quests to find them.
Teamplay is also likely - you could kick around in groups like Ash & co, and perhaps fight evil teams.