As long as the plant isn't diseased, you don't ruin the soil, you don't uproot the plant (which could damage the other plants' roots), etc. the death of the plant should allow the other plants to grow larger and produce better (due to the increased space and nutrient availability)—unless the rotting dead roots harbor bad fungi or something.
I've had multiple plants per pot many times, and the death of one has always allowed the others to get bigger. I've never had issues with it harming other plants, even with the roots of that one rotting (not to say it doesn't happen). I just had them in those containers pre-transplant (but I did let some grow for months that way—mostly tomatoes, but other things, too, including peppers).
Peppers have sensitive roots—much more sensitive than tomatoes. You don't want to uproot the old one unless you don't mind waiting potentially a long time for the others to recover (snipping it off with scissors is a better idea, IMO and experience). Tomatoes can recover pretty fast, though (like in a few days). Peppers can take weeks or months to recover sometimes, depending, but they may not always take so long.