Is it ever a good idea to split-push? I mean, every time there's a chance someone gets caught and it ends up being a 4v5 team fight.
Is there ever a justifiable reason to split-push (or backdoor) in a game?
There are a few good times to backdoor/split push, for example, Yi is an excellent back-door champ because of his insane AS, or champions like Shen and Gangplank who's global ultimates allow for them to instantly access the fight. Another thing you can do is get teleport, so when a fight happens you can get there quick.
When I play champions I split push with (namely udyr), I usually wait until I can 1v1 anyone on their team so it becomes an advantage. If my team doesn't fight and they send someone up to take me out, I kill them and we win the fight. If they send two to get me, my team forces 4v3 and I run away. Either way I get farmed and it's a win-win.
Every time you can get away with it.
Split-pushing and backdooring are agressive strategies that force the enemy team to make a choice: Lose a lane to a split-push/backdoor (5man team push against the rest of your team) or defend the split-push/backdoor and split up their team. Forcing your opponent to make a choice leaves them with the ability to make the WRONG choice which is in your best interest. However this is only advisable when you are confident you know all or most (or the most probable) of the outcomes the choice leads to. Provided you are aware of the gamestate and are watching your minimap (wards help) split-pushing/backdooring can be very advantageous but the strategy also has some risk.
Instead of asking "When should I split-push?" I think we should look at when you should stay with your team.
Insufficient warding
If you can't see any of the enemy team you probably shouldn't even be on their side of the map (in a split-pushing fashion) alone. You need to know that you aren't going to get melted by 2-5 members of the enemy team. Dying while starting a split-push leaves your team down a champion. 5v4s are never advantageous and you are potentially giving up control of Baron, Dragon and your towers if your 4 teammates cannot defend from a team push/teamfight.
Enemy is going to overpower your 4 teammates under a tower too easily
A group of 4 can defend against 5 under a tower to varying degrees of success. If the enemy team has too much burst damage and/or tower diving capability, you may find your 4 teammates unable to defend the base. This means that while you slowly take out creep waves and towers (slow relative to a 5 man team taking out tower after tower) you may find that you don't have a base to return to when you are done pushing.
Your 5-man team is already winning all the team fights
Instead of splitting up and potentially getting picked off stick with what works. If your team scores an ace (kills all 5 enemy team members at the same time) you open up the potential to win the game right there.
Conclusion
The objective of the game is to get to the enemy base and kill it. Towers and inhibitors stand in the way of that. Every chance you get to take out a tower at no cost (no allied deaths, no lost towers, not giving up baron, etc.) you should be doing it. Just be conscious of when the risk is greater than the reward.
Generally not, right? Against a competent team, you generally want to stick together to maximize survivability. If someone on your team is fed enough to solo push effectively, by extension of the fact that accruing gold is somewhat a zero-sum game, their solo adventures will probably doom your team to a pretty sad defeat.
OTOH, if you're playing against a disorganized team, it's quite often the case that 4 players could distract the other team for long enough a period of time for your pusher to basically win the day. That said, why would you need to do this, unless you're playing for the lulz?
Like all things in this game, just use common sense.
The question can be answer by solving the reverse condition: when is it bad to backdoor.
You want to avoid at all cost being aced. If leaving your team may get them in danger of being wiped and allow the enemy to get you or force you back, you will probably lose more than you gained. Especially late game because respawn timer are so long, allowing the enemy team to push more than one tower. Of course, if you're so fed that you can take them all by yourself, why bother?
If your team does not need you to survive, then you should probably go for it.
If the situation is not so clear, you should have the certainty of either:
This requires knowledge of your champions capabilities and map control.
Split pushing is a viable strategy. It does however require a great coordination with your team. Having good map control is also quite important, so I recommend warding their jungle, since you'll be overextending quite a bit.
Usually, champs that have either global presence or high mobility excel at split pushing. Some examples would be:
If you're playing solo queue, this sort of play might be rather risky since your teammates tend to be rather unpredictable. If you're playing a with friends or a premade team, then this sort of strategy is definitely viable — it just takes a lot of practice.
Yi is imo one of the best split pushers. He is fast (so he can switch between lanes), he has good damage, and mid to late game he can pretty much kill any champ 1 vs 1 (exhaust helps).
On top of that his ulti allows him to either burn down a turret even with 1-3 champions attacking (remember turret is worth around 3 kills for team) him or the more common scenario run away when you see 3 coming for you and turret still has a lot of hp left.
The ulti has a short CD so it should be use often. Just watch out for stuns/snares which are his weakness. Also even after you kill a turret, it's sometime better to just use your ulti to kite 2-3 members of the other team around by running around the in the jungle or into their base just to waste their time while your team pushes their lane.
I have an average stats around 6.6/ 6.4/4.4 when I play Yi, but I average around 2 dragons and 2.88 turrets (that is counting even those horrible losses when you just can't do anything). So I'm avoiding as many fight as I can just to push lanes, cause they can't push back as fast as Yi can ^^
Now is people just could stop suiciding defending in the middle of nowhere when I'm split pushing... that would be great...