I think this would be useful in making a magic item. Flame Arrow Quiver: Once per day you can speak the command word and the arrows in the quiver deal 1d6 fire damage in addition to any other damage they would normally deal. Still not awesome, but I'd take it.

A lot of the EEPC spells are whack, just something different about their wording and general balance. To make this spell more favourable I'd happily make it a bonus action (like most buff spells that have no direct effect when cast), remove concentration, and have its damage scale by 1 additional damage die for every 2 spell slot levels starting at 4th (2d6 fire damage with a 4th or 5th level slot). It would still be limited by the number of ammunition, which I feel is fair for having no concentration.

In this case it works as a small damage buff for your bonus action and keeps your concentration free, it won't replace hunter's mark since that's a lot cheaper with a 1st level slot, and could be used alongside it, requiring 2 bonus actions to set up. It's competing with the likes of elemental weapon; which is more diverse with its choice of damage types and provides an arguably better damage boost with its bonus to hit that also scales with spell level while having no limit on number of attacks, and holy weapon which is in the same boat, is a bonus action, but is also a 5th level spell base, so should be better than an upscaled flame arrows.

For the many people that state that Hunter's Mark is better... This spells isint just an extra 1d6 fire damage... It also sets the target on Fire. Which means they take an extra 1d4 fire damage on the start of each turn if they do not extinguish it. As a Ranger your role is to whittle the enemy down not one shot them. So as a 3rd level spell I think it's pretty decent. 

It says touch a quiver, but it’s not listed as a material component so I say it works. 

would  a fire arrow work under water??

I have an idea for this with a kobold alchemist artificer. As a kobold artificer, the character is obsessed with making traps. I imagine the character makes a bag of holding at level 2 and over their career has collected every heavy crossbow they come across until eventually owning 12 in their bag. 

At least once, I'd try to get into a situation where he could build a trap positioning all 12 crossbows to be pointed at a pressure plate that once stepped on fires all 12 bolts at once. Each crossbow would be loaded with a bolt enchanted by this spell.

Each bolt would roll to hit (modifier to hit based on DM and probably influenced by the skill check to build the trap in the first place). 

This trap would deal out 12d12 + 12d6 if every bolt hit. 

Nowhere does it say it "sets target on fire." Spells do what they say they do. Dragon turtle's breath weapon of hot water does fire damage but i can guarantee the hot water doesn't set targets on fire.

Not all spells are created equally.

It's better to rule this as a non concentration spell, and give it an unlimited duration (or just a day) for 3rd level if you wanna incentivise your ranger getting it, because they'll most likely cast it at night to boost some arrows they have. And if they're casting it during the day, they're wasting a 3rd lvl slot that can be useful elsewhere. Totally overshadowed by Hunter's mark otherwise when it does the same thing and cheaper.

"arrows or bolts"

Quote from BirdMonk >>

Can this be used on firearms?

 "ARROWS OR BOLTS"

I'm surprised that no one pointed out that this can be used with Hunters's Volley multiattack skill. its not a huge buff, but if you know there's an area with saturated targets and can hit 3 or more targets it's certainly useful... now because rangers have to choose spells they know rather than preparing them like a druid... it's still a bad choice for a level 3 spell

Quote from dylanfooley2000 >>

For the many people that state that Hunter's Mark is better... This spells isint just an extra 1d6 fire damage... It also sets the target on Fire. Which means they take an extra 1d4 fire damage on the start of each turn if they do not extinguish it. As a Ranger your role is to whittle the enemy down not one shot them. So as a 3rd level spell I think it's pretty decent. 

 It doesn't do that though. The spell does exactly what it says it does, and "on fire" isn't a universal condition in DnD 5e. 

Sure you can use this with Volley, but that still takes an action to set up, and you could just use the spell Conjure Volley, which deals magical weapon damage, far more reliable than fire damage.
Rangers get decent low level spells and iffy 3rd level ones...

It's hard to think of a scenario where this spell isn't F tier.