I’ve downloaded a couple adventures on my iPhone for offline play but when I open them they say my session has expired. Do I need to download a specific software to play them or is it just not possible on mobile?

To play Quest games, you need the Quest software, which currently only runs on Windows.
If you want to play on your phone, you will need to use the copy of Quest installed on this website's servers, which means you need to be online.

Do you know if there will ever be a mobile version?

I think the developer of Quest Semi-retired after the last update. But thankfully some of the veterans here seem to have a way to get in touch with whomever maintains the site/servers if there are tech issues.

It would be really neat to at least have a mobile version of the player, then people could publish games and put them in App stores for free or small fees. But I think app development and maintaining them with versions for the Apple and Android updates is a LOT! of work.

When I watched the documentary that's linked to from somewhere on the site, I thought. "If they want these marketable to the right audience, they need Amazon to want a supportable platform for it running through their Kindle, like audible for audio-books. It would be smart to have a "Quest" for Kindle.

People already have the type of IF that are the choice books in appstores, but I think even those aren't on Kindle. Or are they?

Still, if there's an app developer out there in the forum, it wouldn't be a horrible idea!