At what point do you want to start pushing instead of just last hitting? Is there a reason to do it at all after any towers are down?
You never stop to try and last hit the minions, except when you're playing support; then you'll leave the last hitting to the carry you are with. But at times it's good to push minion waves to the enemy tower (either using your spells or starting to auto attack), since enemy towers can and will deny gold (and XP - only when your opponent isn't there) from your opponent. But you always have to look at the map when you are going to push, because overextending and dying to a gank is worse then letting your lane opponent farm.
Conversely, if the enemy is pushing his minions closer to your tower, you can start auto attacking the minions to get them down faster, but even while auto attacking you still need to watch minion hp so you can get the last hit.
Always last hit; you want the gold! Even when pushing a lane, time your attacks so you get the last hit.
I think your question could be rephrased as "When to push, when not to push?" In general, it's risky to push a lane because you're left in a vulnerable position. The two biggest reasons I push early are to heal with lifesteal, or because I'm beating my opponents in lane. It's risky, but an early tower break provides a nice boost for your team and frees you to float more.
Advantages to pushing:
- Tower damage
- Deny opponents exp/gold
- Healing with lifesteal
Disadvantages to pushing:
- Vulnerable position
- Uses resources
Some tips while pushing:
- Keep wards up in the river (you don't want to be ganked)
- Watch for MIAs
- Use care if an enemy champion has taunt. Taunt + turret = pain
tl;dr: Only push if healing via lifesteal or obviously winning, and then make sure you ward any potential gank spots.
There are a few times you should push.
- when you are close to an enemy tower and the enemy needs to recall
- when ad carry and last hitting goes well you want to be at their tower at around 8 to 9 minutes so you can recall and get your B.F sword
- around 15 minutes if you are winning and you can push down the enemy tower safely
After the 15 minutes push you should go to mid to get that tower down. Otherwise you should try to just last hit. Also if the enemy recalls and you're far away from their tower only last hit so they lose more experience.
The only time you really should JUST be last-hitting is when your enemy clearly has no means of killing minions while you're laning. Some examples of this would be if you're up against two melee champions and you have two ranged. If you keep them off of the wave then stand between them and the wave combat and just deny them EXP and gold then thats a good time to last-hit. Also if your in a 2v1 lane (you being the 2) you should last hit until you can force the solo to recall then push. A great example of someone you should be last hitting against is Tryndamere. He is worthless early game in the waves so if you can keep him out of the combat by griefing him then you can last hit him out of a lot of gold and exp if you do it properly.
I don't know why these people are saying not to push and last hit. If you have the power and you're not putting the enemy at a disadtvantage by last-hitting then push. An enemy behind towers is a lot easier to deal with then an enemy in the brush or MIA. Keep your enemy pushed back to their tower and their tower will steal their gold. Towers mean everything in LoL, not worth your life, but taking enemy towers greatly decrease their map awareness and their map control.
Edit: Also I wasn't implying that if a minion is low on health you shouldn't focus it to get gold, I was implying you shouldnt JUST last hit minions except in these cases.If a minion clearly has low health then kill it, but that is just common sense.