Stewart is recovering from a broken right leg suffered in a sprint car crash Aug. 5 at Southern Iowa Speedway in Oskaloosa, Iowa.
The 23-year-old Dillon, who won the NASCAR Camping World Truck Series title two years ago, is competing in the Nationwide Series this season and holds the points lead.
It will be Dillon’s 10th Sprint Cup start.
Max Papis drove the No. 14 to a 15th-place finish Sunday at Watkins Glen International.
SHR has not determined who will drive the car at Bristol on Aug. 24.
No timetable has been established for Stewart’s return, but the three-time Cup champion was discharged from the hospital late Sunday afternoon and resting at home.
“He’s doing well,” said Greg Zipadelli, competition director at SHR. “As far as when he’ll be back, I think we’ll know mid-week a little bit more on his schedule. He was doing as good as he could. We did talk about what we had for plans there, and he was all on board and felt like we had made good choices. So that was important.”
When asked if Stewart might possibly get in the car for a lap or two at future races just to get the points and remain in the hunt for a spot in the Chase for the Sprint Cup championship, Zipadelli said the team would leave that up to the medical experts. Stewart dropped from 11th to 17th in points after Sunday’s race.
“Honestly, if the doctor said we could do that, we’d probably do it,” Zipadelli said. “We’re going to do everything that they say because we want him back a 100 percent. Sometimes you can do things sooner than you’re supposed to and make your healing process longer.
“So we’ll follow on with what the doctor says. If the doctor says in four weeks or six weeks he can do that, then we’ll probably look at those options and do what we can do.”
Zipadelli said SHR had heard from about 50 drivers who were interested in replacing Stewart until he’s well, including “people that I didn’t even know raced.”
The goal this weekend, Zipadelli said, is to run all the laps.
“We need to finish the race,” he said. “Right now we’re still fighting for all the owner points we can get, so finishing these races is very important. I’m not saying it won’t change, but right now, it’s no different than if Tony was in here. We’re not treating it any different.”