some may, but the majority no.
Most stay because they really believe its the truth.
Some may be disillusioned, but have no real alternative. That can really suck at times. The deeper you're in, the harder you can get out.
For an elder, even to quit his responsibilities,this brings up several aspects:
1) questions why ? What reason can ge give with what consequences ? If he says it is stress related, what other elders will offer help to relieve stress, but at the same time pressure him to remain. Is it health, well, its no reason to quit. Is some bad conduct ? Well further investigations are being made. Worst of all, does he have doubts ? Is he spiritual weak or even borderline apostate ?
2) Rumors will start, regardless, since the reasons for quiting are not announced. Family will suffer to some extent. Especially if the elder was known as liberal and open minded, this might backfire too.
3) His position to help others, especially family, to think about certain matters are being diminished. He even might become "bad association" sooner as you can say "governing body"
All this would not matter, if he is single and has no family in the "truth", then he could pull out easily, perhaps loosing his social envrionment, but able to build up a new one. This might be also difficult, but its more worth it than loosing his family or marriage too.
However, it all comes down to ones personal situation anyway.