I am not familiar with the 1991 230E model, and just acquired a 1988 300E and to date have had little or no's side problems rop up. But my 1986 190E 2.3-16, which did not have a keyless entry system installed, had the central locking system de once the car got a few years on it due to corrosion of the wiring under the driver side floor mat. This made the central locking system fail and I could only open the front doors or the trunk with the key. The problem lies in water collecting under the carpet (the wire in question is routed on a 190E like mine from the distribution bus connector just inside the driver's side side kick panel to left of the clutch foot rest, then under the carpeting along the floor to the right rear passenger seat where the little box with the vacuum pump and various connections from the door and trunk locks is located). In my case the wire was chafed and water got in there (I assumed at the time it was from the snow and water that came in on my feet in the winter, but now believe it might have been from clogged sunroof drains or the drains around the hinges of the hood, as others in this forum have suggested) and in a period of years the copper turned to blue-green powder for about an inch or so. I replaced the length of wire with good grade stuff and carefully sealed the splices. No more problems for about 8 years.
Anyway, I hope this gives you a place to start as when I read the doors don't unlock once you are in the car I began to think maybe the issue is not with the remote system but the onboard central locking system. I believe you also have a small vacuum pump under the rear passenger seat. To test whether or not this unit is getting power, run a set of wires to the unit from the battery and if the locks then work, you have found the cause. Finding the site of the electrical discontinuity is going to be more difficult, but the problem should be similar in that the wiring along the floor boards is subject to various forms of moisture and other stresses. Hope this helps, Jim
1986 Euro 190E 2.3-16 (291,000 miles),
1998 E300D TurboDiesel, 231,000 miles -purchased with 45,000,
1988 300E 5-speed 252,000 miles,
1983 240D 4-speed, purchased w/136,000, now with 222,000 miles.
2009 ML320CDI Bluetec, 89,000 miles
1971 220D (250,000 miles plus, sold to father-in-law),
1975 240D (245,000 miles - died of body rot),
1991 350SD (176,560 miles, weakest Benz I have owned),
1999 C230 Sport (45,400 miles),
1982 240D (321,000 miles, put to sleep)