I have 7/8 inch painted old growth plank flooring (pine? elm?) with no sub-floor. I would like to sand and refinish with poly. How much material can I remove and still have a strong enough floor to support routine use?

Old flooring is milled with the tongue in the center of the thickness. The tongue will be what supports the floor. The top part of the groove will be what is the weak part when it gets sanded. Usually the floor is nailed down with 8d box nails. When you reach the top edge of the nails in a few spots the floor will be getting weaker so that there can be splinters from the top part of the groove cracking away from the weight of a large person walking or running over it. To help strengthen the floor you may be able to add blocking between the floor joists from the underside if it is accessible even if nail heads start to show from being sanded a lot.
There will not be a safe thickness, it depends on the traffic over it. A thin floor can last a long time if it is not stressed by the load.
A strip of plywood glued to the bottom of the floor in between the floor joists will dramatically stiffen and help support the floor and lengthen it's life as it gets sanded thinner.

Thanks Pete A.! I also have some old bug damage so I may have to use both plywood and blocking to sleep at night.

I have used epoxy wised with wood flour to fill termite or beetle damage after the tubes have been exposed. To match an antique pine color I will use red oak edger dust. The flour will darken when mixed with two part epoxy.