Identify your goal, specifically, and then isolate the obstacles to that goal.
Goal: accurate placement of furniture during performance.
Obstacle: spike tape comes off of floor surface.
Response: staple spike tape? Result- spike tape comes up
Response: use sharpie marks? Result - unkown. How about trying a sharpie mark in a corner of the carpet (like testing the colorfastness of a garment) and seeing if it actually damages the look of the carpet, or if it will come out with a cleaning solution like GooGone or something?
Response: Carpet tacks & tacking hammer? Result - unknown (good suggestion, mc)
Response: Train crew to LIFT furniture rather than dragging it?
Response: Marly tape over the spike marks? if the carpet is a low pile stiff carpet, maybe this will work
If all the responses fizzle out, REVIEW YOUR GOAL:
just how necessary is accurate placement of furniture? does furniture move a lot? Do you have a lot of lighting specials focused specifically on the actors on this furniture, or hard focus cuts around it? Are spike marks the only way to gauge where to place it - could you reference off of set pieces or corners of the deck, or other pieces of furniture that haven't moved yet?
Don't assume that you have to spike mark everything just because stage managers always spike mark everything. Spiking isn't the goal. Accurate, repeatable performances are the goal - and if your tolerances are wide enough, you might not need spike marks at all.
If placement really must be precise, and spike marks are coming up during a run (so there's no time for you to re-measure and replace them) then this is a TD/Producer issue, and I would get permission to disfigure the carpet, or get a replacement carpet for the set, or lower the standards of accurate placement by getting the lighting designer to make changes (for example)