Saudi vision 2030, which is a nationwide vision of the future of Saudi Arabia mainly about finding other sources of income beside oil, has tourism as one of the items. It's coming for sure but when, no one knows yet. I have spent some time trying to get an expected date for this, with no luck.
All the official sources just list it as one of the items. Implementation of the vision items are mandatory, some of these items will be sooner than the rest, tourism is not listed as one of those to come soon. So, as of now, the only date that we can be sure of is the 1st of Jan, 2030, which is the date in which all vision items must be implemented.
Anyway, many steps have been taken lately that indicate that this is coming sooner than 2030, one of them is removing all authorities from the religious police, which is in my opinion a very important step and the thing that was holding back this idea since the beginning. Saudi people in general are welcoming people, they travel a lot and they would love to see people come and visit their country. Another step is the start of implementing a program similar to the "Green Card" program, the current permanent residents started to get a new card called "Resident ID" valid for 5 years, the old "Iqama" card which is valid for one year is no longer issued. All these small steps are in my opinion leading to making Saudi Arabia an open country. IMO, another Dubai to be born as of 2030.