The NWS in Seattle is predicting snow starting early next week, see Web page, probably down to 3,500 foot elevation, which is much of Mt. Rainier NP. All the entrances are at 2,000-2,400 feet elevation and while the river valleys continue to be below the snow prediction, all the rest of the higher elevations will have snow.
If this happens, the NPS will be ready to implement winter snow rules for the NP, such as closing the White River road, Mowich Lake road and the Stevens Canyon road at the entrances earlier than normally scheduled on November 1st. If this happens, it's unlikely the entrance will reopen as the NPS will just leave them closed until the spring.
This also means the NPS may implement the road closure for the road from Longmire to Paradise nightly at the gate just east of Longmire. The road won't open in the in the morning until after the parking lot and road to Longmire from Paradise is checked and cleared of snow, which is usually 8-9 am. And there is no overnight parking at Paradise without a camping permit (available at Longmire) or camping in vehicles of any type (cars, trucks, RV's, etc.).
The good news is that this is common as the first snow storms, and is usually short-lived. Warmer weather usually follows these storms and the snow melts below 5-6,000 feet until the normal seasonal snow in November-December period.