It'd make them more balanced with other races
Detect Balance is a fairly popular guide for homebrewing races and determining whether they are balanced with one another. It includes scores for the official races and breakdowns of how those scores are determined. It also assigns values for various possible racial features.
The second tab of the sheet provides some info about the sheet and how it's meant to be used:
The average score for PHB and EE races by this scale is about 25.
The recommended score for new homebrew races is 24 to 27. Races should generally not eclipse 30 or fall below 20.
Values are approximate. If a race is few points higher than another that's not statistically significant. It doesn't mean it is better. Depending on circumstances it may not perform as well. Every campaign is different, what turns out to be useful will vary a lot.
The scoring unit is in quarters of ASIs, or Ability Score Increases. In other words a feature worth 4 is approximately equal in value as being able to raise an attribute by 1.
The third tab of the sheet has a graph comparing the official races at the top, followed by the individual breakdowns for each race. Dragonborn are listed first, being alphabetically the first of the PHB races. They're given an overall rating of 21 points. Their racial ability score increases of +2 (to Strength) and +1 (to Charisma), totaling +3, are obviously given a rating of 12 points. Their breath weapon is rated as 6 points and their damage resistance is noted as being worth 3 points, with the note:
The breath weapon and resistance are nicely balanced-- If you pick rarely resisted damage for your breath weapon, you likely have a resistance to an uncommon damage type.
These ratings generally match those described on the first tab of the sheet.
Now, let's look at the point ratings assigned to darkvision on the first sheet. 30 feet of darkvision is given a rating of 2 points, 60 feet of darkvision is worth 3 points, and 120 feet of darkvision is 4 points. If we add the typical 60 feet of regular darkvision to dragonborn, this brings their total point rating up to 24 points - still well in the middle of the usual range for most races.
By comparison, the various dwarf subraces (plus base race) have 29-30 points; the elves have 27-28; gnomes and halflings have 23-24; half-elves have 29; half-orcs have 24; and tieflings have 23 points. (Humans are outliers; regular humans have 17 points, while variant humans have 33 points.)
Dragonborn are normally on the lower end of the typical range, and adding darkvision brings them back to about average.
(Of course, any such rating doesn't mean much in a vacuum; certain low-rated races might be better for certain builds, and certain highly-rated races might be worse for other builds.)
Best Answer
This seems like a fairly substantial change that would have a pretty dramatic effect on game balance. Consider that this grants elves resistance to wooden weapons like clubs and quarterstaves as well as natural weapons. Also bear in mind that it isn't just animals that use teeth, claws, and fists - you're also talking about elementals, some angels or demons, aberrations, and a lot of other creatures. I would say that you're potentially halving the damage that elves would take from a pretty significant number of enemies.
Now, obviously you can mitigate that effect quite substantially if you're planning out all of the encounters yourself by ensuring that your elf PCs very rarely encounter enemies who don't wield steel weapons. Consider, though, that if the rule change has a very limited effect on gameplay it might also fail to generate the flavor that you're looking for.
One way that you might be able to generate the desired flavor without having much of an effect on gameplay is if the elves themselves use primarily wooden weapons to avoid having them turned against them. In other words, most people know that they need to carry metal weapons to overcome the natural resistance that elves possess, and magical beasts like dragons or demons automatically penetrate that resistance, but the elves themselves use primarily wooden arrows as a way of minimizing the damage from friendly fire, or wooden quarterstaves in case they're disarmed or otherwise have their weapons stolen.