By 'thrust-firing' I mean moving forwards as you fire, if only for a moment. This forward thrusting adds to the shot speed, and with at least some of the weapons it can make you significantly more deadly.
Great with the needler
The needler is the best example for using this tactic. If you're running forwards as you fire it, the needles travel noticeably faster. Presumably, those clever Bungie folk have done their physics homework and made sure that velocity is additive. The result is that enemies have less time to dodge, and an explosive succession of needles will enter them more rapidly. Maybe it also focuses the aim better; it feels like it, but it's hard to test. Anyway, try out some aggressive thrust-firing with the needler and I think you'll soon feel the difference.
Faster plasma balls
Velocity is also noticeably additive when firing a plasma ball from a plasma pistol. You can easily check it by counting seconds before it hits a distant cliff. I'm not sure it'll do greater damage than usual though. But regardless of any increase in damage, the enemy has less time to get out of the way, just as with the needler.
I also use thrust-firing with most of the other weapons. With the pistol I use it at medium to long range, and I'm convinced that the Elites in particular don't like it. They seem to go down like a ton of bricks sometimes, when I manage to slam in the headshots. With the plasma rifle I use aggressive thrust-firing (in short pulses) into the heads of Elites, which tends to also have a stun effect that grants you more time. Similar thrust-firing into the exposed areas of advancing Jackals is also highly recommended. With the rocket launcher, thrust-firing reduces the flight time, which can help when going for a moving target; but I don't know whether the rocket will cause more damage.
Using it with the Banshee
Thrust-firing can also be used with the Banshee's fuel rod cannon to make the shot fly straighter. It aids accuracy and helps you keep your distance. As you hover, briefly thrust forwards with each shot. While the cannon is recharging, you've got time to regain any altitude lost.