EDIT: Okay, so lots of these are Do's and not Don'ts. It's 3 in the morning and I'm not gonna rewrite them, ignore them if you want.
If you're squishy and fast, don't rush to Top without your team backing you up--at least, unless you know their whole team is on their mid (because you're smart and watched the map). Wait for your team to catch up or else you'll die for nothing and your team will be weaker when they get to the Windmill.
If Twitch (or another stealther) is ruining your fun, buy an Oracle's. Or better yet, a Sweeper if you're AP.
If you're bot-laning, don't go into the brush. The enemy will see you do it and brush won't help you against Anivia's ult or Sivir/Ahri/Xerath (etc) with their skillshot nukes. Hug top. The fog of war will hide you much better and there's a tasty health pack up there with two much more effective brushes too.
Playing stealthers, and you can't see them on the map? That enemy mid looks tasty? The stealther is often there waiting for you.
Pay attention to the map. Most players go straight in the direction they're headed, so even when they get into the Fog you know where they're going (but see below).
Abuse the fog. Most players don't pay attention so you can use your point's fog to ambush them. There's a line from your base to the center fog that will hide you if enemies aren't on your point. Why shout out and let them know you're back? Why tell them where you're going?
Keep abusing the fog. A smart player will be watching the map and may well see what direction you're headed. So go somewhere else. Fake them out. It costs like one second of movement to lead them in the wrong direction.
Those minions cap faster than you, help you cap faster, and can't be interrupted by anything short of CC or death. If your team is busy elsewhere and you have an AoE (Rake, Boomerang, Thunderclaws, etc) clear a wave or two to help you cap especially if you suspect a stealther is there or Shaco left his box. DON'T farm minions if two of your allies are already capping the enemy point!
The Tab button is your friend. Make love to your Tab. Frequently.
It's usually worth it to wait for your team.
Don't chase an enemy to get that kill and ignore that undefended point. Run back to cap the point. If they come back, use your newly-refreshed cooldowns to ruin their day.
Don't use Clarity on Dom. Ever. Just no.
Don't whine about someone stealing your kills. It doesn't matter who kills them as long as they're dead. It's not Farmer's Rift, you don't need the gold for your ubercarry. You don't need the points for your epeen score. You just need them dead. Sure, you had the enemy on the ropes and were about to strike a killing blow when Talon comes out unecessarily and disassembles your target. Say thank you so you can both team up on the point's defenders!
Don't feed the trolls. Negative whiners who complain more than they play aren't worth arguing with. Even if you have three AFKs, or it's your first Dom game and you utterly suck. Ignore them and learn as best you can. The game isn't lost until your Nexus health reaches Zero, and we've all seen that win with one health point remaining because you pulled it together long enough to beat them.