Novice Engineer is a 1/1 creature, costs 2 mana and allows the player to draw a card when it is played. Under what circumstances (if any) would it be worth playing the coin on the first turn to summon this card, sacrificing future mana advantage for immediate card advantage?
In almost all situations it would be best to save the coin. Assuming every other card in your hand cost 3+, it still allows you to slightly beat the tempo with one of those cards and to play your Engineer or a "3 drop" on turn two.
Other factors to consider, besides your hand, is your and your opponent's class abilities. If you are playing against a Mage or Druid, they could easily use their ability turn 2 to kill off your creature and you have basically exchanged you 1/1 and the coin for another card on turn 1 (making this a horrible play). Even if it was a rogue, getting in 1 damage would not really be worth the cost.
If you are running a Tempo or Rush style deck such that the majority of the cards are 1 to 3 cost and RNGesus gave you all of your 4+ cost cards, I would still save the Novice Engineer for turn 2 and play a 4 cost minion on turn 3.
Generally speaking, it is usually a bad idea to use the coin on turn 1. However, if you have another good "2 drop" in your hand, that would be the only time I would consider using the coin turn 1. For example, if you use the coin to drop a Faerie Dragon turn 1, you could then play out your Novice Engineer on turn 2.
I am stretching REALLY FAR here to come up with any other ideas where it could be an acceptable play. If your opponent had an ability that would not affect the field, such as a priest, it might not be a horrible play (still, depending on your hand there are probably other uses of the coin that are better).
It depends on a couple things first. Your hand, what hero you are playing against and what type of deck you are playing. I'll put them in order of importance to me.
Your hand: If you mulled and got a bunch of higher cost cards, it's worth coining. You want to cycle through your deck as soon as possible to get some cards you can play in turn 2, 3, 4. Not having anything to play on a turn is really bad. It's worth it in this case.
If you are playing against any type of rush down deck, DO IT. Even a 1/1 creature can help slow down their tempo. Also that extra card could help if it's a removal card for some of their mid-tier cards. If you are playing against a late-game control Druid/Priest for example, it's not worth it. A 1/1 creature or even another card will not help in turn 2, 3, or 4. Save your coin for a higher cost card that will speed up your late game.
If you are playing a Paladin, it might be worth it. You could buff your NE by giving it divine shield, extra attack or add another 1/1 soldier. If you're a druid you could give it +2/2 and taunt. If you are playing a high tempo deck, it might be worth it to hit them for 1 damage on turn 2 or even make a Mage use their hero power to end her turn 2.
More times than not, it's worth it. But again, it all depends on a few factors.