My Mum has a similar problem, either leaving the handset of an ordinary telephone not quite in the right position in its cradle, or forgetting to push the "hang up" button on a cordless one.
I can think of two solutions, but have not yet got myself to implement either:
1) a quick, dirty and possibly a tad expensive solution is to have a mobile phone somewhere in her home, that will not be used for talking, just as a "remote ringer" to alert your/my mother that the regular phone needs to be put in order. We would need to take care that the battery of this mobile phone does not run flat; perhaps we can leave the charger permanently connected, or plugged into a programmable time switch that will turn it on once a week for a few hours.
My Mum recently got a mobile she can actually use (another area where modern technology leaves the elderly out in the cold), so perhaps the problem will go away.
2) A more elegant and effective solution is to use an automatic device connected to the telephone line, that will beep or ring loudly if the phone is left open for longer than, say, 45 minutes. Ideally, the device should detect the presence of voice in the line and stay calm if Mum is talking (no matter how long), and only raise alarm if the phone is open with no conversation going on.
The problem with solution 2) is that I have yet to find such a device anywhere on the global market, or find the time to build one myself. I don't even know what it would be called, so it is difficult to run a web search. If and when I do build it, it will be illegal to connect it to the telephone line because it will not be homologated, but I have no qualms about that. Better be judged by twelve than carried by six.
Hope this helps,
Leszek