This is frequently debated, but really there is no firm answer.
The feat just doesn’t explain this point, and it’s relevant. The extra damage from a Dragonfire Strike is fire damage, which may well imply that it is not precision damage. Or you could argue that there is no reason that precision fire damage couldn’t be a thing. Or you could argue that, if you wouldn’t have gotten sneak attack damage because the target was immune to precision damage, then Dragonfire Strike doesn’t trigger at all as there is no sneak attack damage to convert to fire damage.
All of these arguments have merit and are plausible interpretations of the rules. It’s really impossible to authoritatively state that one of them is the correct interpretation of the text.
Expanding a rogue’s ability to sneak attack would be good for the game, however.
The rogue is not a high-power class. No class that gets sneak attack, sudden strike, skirmish, or other forms of precision damage is.1 Precision damage has a ton of limitations, and does not result in particularly-high damage totals (Power Attack and charging offer far more). When Dungeonscape offered the opportunity to let rogues get half of their sneak attack damage against things otherwise immune, in exchange for evasion, that was widely seen as a must-take trade. Carrying wands of golemstrike, gravestrike, and vinestrike, spells from Spell Compendium that allow sneak attacking constructs, undead, and plants, respectively, is widely recommended, despite the costs and awkwardness.
Meanwhile, feats are phenomenally powerful things, and you get all too few of them. The very-low power level of most feats often leads to them being undervalued, but if you ignore the 95% of feats that aren’t worth taking, some of the things you can do with feats are amazing. The Craven feat from Champions of Ruin increases a single-classed rogue’s sneak attack damage by more than 50%! (And for multiclassed rogues, it can be a much higher percentage than that, since it’s based on character level.) (Admittedly, Craven has a rather-substantial downside, too, at least at higher levels of the game.)
So the long and short of it is that spending a feat for Dragonfire Strike in order to get an option that allows you to sneak attack some creatures you otherwise couldn’t is good for the game. Rogues et al. desperately need options like this, because they are underpowered. It’s problematic in that everyone who wants to sneak attack is all-but-forced to take the feat, but that problem is distinctly smaller than the general problem of sneak attack being substantially underpowered.
- The unseen seer prestige class from Complete Mage is pretty good, but the precision damage bonus is not the reason why: the combination of 6+Int skills, ¾ BAB, and 10/10 spellcasting is the reason why.