Thank you for the theological question for Premillennialists, “Do eschatological futurists (premillennialist) think the only test from God they need to worry about is a mark of the beast?”
Premillennialists believe that Christians will be rapture by the return of Jesus before the Great Tribulation begins. They will return with Jesus at the end of the Great Tribulation known as the Marriage Supper of the Lamb. The mark of the beast occurs during the Great Tribulation. Thus there is no test since Christians will be absent from Earth during the Great Tribulation.
Postmillennialists believe that Christians will go through the Great Tribulation and need to avoid the mark of the beast. They believe that God will enable the believers to persevere through the persecution or even die for their faith. When Jesus Christ returns at the end of the Great Tribulation, they will be taken to be with him and return to judge the earth.
If there is a test for Christians, it is whether Christians will continue to believe in the return of Jesus Christ whereby they want to live godly lives for God or abandon that belief whereby they will adopt the values of the world.
The belief in the return of Jesus Christ is no longer a central tenet of Christians. It has become eat, drink, and be merry for Jesus Christ will not return.
SUMMARY: Postmillennialist have a greater concern for the mark of the beast than Premillennialist.