Many franchises have done very well with different actors in the same role. It would take a while to adjust to the new actor, but I suspect thirty minutes in or so, I'd be just fine. Ultimately it's the story that draws me in.

I think the article summed it up nicely;

There are so many stories to tell, so many points in history to visit, so many treasures to unearth that continuing the adventures of Dr. Henry Jones Junior isn’t just a choice; it’s an obligation. Raiders of the Lost Ark, Temple of Doom, Last Crusade and Kingdom of the Crystal Skull are more than just films; they are, to borrow a line from a galaxy far, far away, a first step into a larger world.

An Indiana Jones 5 would inspire kids to learn more about history, archaeology, and morality, just as its predecessors did, just as cinema should. So dust off the old fedora, sling on the old leather jacket and let the Raiders March play. Let go of your own memories, and let some new ones for new people form, because the man in the hat needs to come back.

I also agree with Tobar and Crow on not making it some sort of reboot\origin. The trend of starting over needs to go away. The audience doesn't need to be treated like a child. Show another Indiana Jones adventure that happens to have someone else playing the part.

James Bond, Dr. Who, Batman, Rooster Cogburn, Darrin Stevens, etc. It's been done before without patronizing the audience with a lengthy and unnecessary explanation and it works well.

I've said it before, I'll say it again; If you can successfully recast 007, Kirk, and Spock (and they did) - you can recast anyone. Granted, the crew of the Enterprise started with an origin story, but it didn't have to. The recasting of the crew was so fantastic, they wouldn't have missed a beat.