Why Games Fail
1 - Single Point of Failure
2 - The GM is overambitious
3 - Something is lost
For me, almost all games I've run have been cooperative/team games, where players can drop in and out freely, so number 1 has almost never been an issue. Usually they've ended through a combination of 2&3; the original spark dies down, whilst the game complexity ramps up, and at some point the effort required to update rises above the motivation available.
I'm not sure how to solve these problems. However, I think you may be looking at things the wrong way, to a degree. The title of the thread is "What makes a good forum game", but then your actual question is "What makes a forum game that reaches a conclusion", which is a different question. Sure, in a perfect world, the best forum games would run until their story was finished. And to be sure, an amazingly awesome forum game that only lasts two turns can leave one feeling rather dissatisfied. But I don't think merely lasting until the end is what makes a good forum game.
To answer the question of "What makes a good forum game", I'd say there are several elements:
-Social: a good forum game engages players not just with the game, but with each other. Otherwise, you might as well be playing a single-player videogame. This can take many forms, from discussing what to do in a suggestion game, collaborating in god games, or direct competition in some cases. This is rarely a problem, but it's worth keeping in mind.
-Organic: a key difference between a forum game and a videogame is that videogames are pre-programmed. Even the most expansive videogame is limited in what it can do, but a forum game can be anything. I'm not saying every game should be a completely freeform mess, as that would obviously suck, but neither should they have rigorous rules and ironclad settings that determine exactly what can happen. This is a matter of opinion, obviously, as there have been many forum games which basically emulate board games, which people have enjoyed, but I prefer more organic ones.
-Engaging: sort of obvious, but players should feel engaged with a forum game. It's hard to pin down what makes a game engaging, of course. There are certain formulas that one can copy for a decent chance at success, but if you want to do something original... well, I don't think there is a way to know in advance whether it will work. Just try it, and be prepared for failure. Don't feel bad about dropping a game that just isn't engaging.
-Flavour: I think most people enjoy a good setting/story, even if it isn't the most important thing. A purely mechanical game, devoid of all flavour, would be strictly worse than one with even a basic backdrop, in my opinion.
-Quality: a tricky element, but a crucial one, is simply good execution. Some people are just better GMs than others, be it by dint of practice or talent. If you're concerned that you may be falling short on this metric, requesting honest, constructive criticism could help.
That's just the things I can think of off the top of my head.
I think if you want to study a really successful forum game, Haspen's SPAMKINGDOM (1 and # specifically) games are very good, well executed, long lasting games, the two mentioned ones of which ended naturally after long runs. There are also SPAMKINGDOM games, as well as various other SPAM-whatever games (by Haspen or others) that
didn't last very long, so studying why some lasted whilst others didn't could help determine what the secret to success is.
The Arms Race genre has many entries, of which several reached a conclusion: the original Arms Race (by Sensei), Wands Race (originally by Iituem, taken over by evictedSaint), and Intercontinental Arms Race (by Sensei again)(although technically the epilogue has yet to happen). There was also another AR by Sensei on a different forum (ICAR is a battle between the winners of Sensei's first two ARs).
I'm not personally acquainted with FEF at all, but I believe several games of that genre have successfully ended? I dunno, ask someone who plays them.
God games basically never end, by nature, as there is no natural conclusion for them to come to- that doesn't mean they aren't fun, of course.
Suggestion games... I dunno, maybe a few have 'ended' properly? The ones with an actual predetermined plot, maybe?