Does a half dragon, with zero racial hit dice get class levels applied to their breath weapon or does it not even get a breath weapon?
The rules state: The breath weapon deals 1d6 hit points of damage per racial HD possessed by the half-dragon.
My GM believes class levels apply. I believe if there is 0 racial HD then there is 0 breath weapon.
Technically, the rules-as-written used to indicate that such a dragon would have a 0d6 breath weapon of the appropriate energy type. This was important because certain abilities can add damage to attacks using a certain energy type or activate when making a certain kind of saving throw successfully. Because the ability specifically calls out 'racial HD' as opposed to just 'HD' the normal HD/class levels equivalency rule most certainly does not apply, so your GM is soundly wrong on that point.
However, James Jacobs, the Paizo creative director, indicated that half-dragons without racial HD should not be given a breath weapon at all here, where he also cited his hatred of half-dragon humanoid characters and fervent desire that people would stop making them as the goal in this decision.
As creative director, James Jacobs is pretty much in charge of the rules, and the pathfinder errata model supports random posts on the Paizo boards made by Paizo staff as a form of authoritative text with errata power, until replaced by a more formalized FAQ post, official errata, or newer printing. Usually if one of these things is planned or considered, the staff member posting the forum post will indicate that the change will be noted in a future source. No such indication was made with the post, so it is presumably intended to stand as authoritative errata on its own.
There are a lot of sources which are almost universally regarded as RAW by among the Pathfinder community, such as the Core Rulebook. There are a lot of sources that are not regarded as RAW equally universally, such as random posts on the Paizo boards by non-Paizo staff members and answer text found on this site. Non-FAQ errata posts are an area of contention, however, as a few groups that play RAW don't consider any material not published in print RAW and many (but still a small minority) groups only consider 'official' errata (these) to be valid.
The point here is that both a 0d6 breath weapon and no breath weapon at all are possible RAW interpretations, depending on your scope for RAW, with the latter being the only correct choice in the pathfinder errata paradigm.
The rules(If your referring to the half-dragon template that has the same wording you used.) seem to indicate that if you don't have racial hit dice you wouldn't get a breath weapon at all. This would also effect anything else that racial HD play a part in like skill points.
A post by James Jacobs(Works at Paizo) confirms this, here
Someone asked:
"A half-dragon retains all the special attacks of the base creature and gains a breath weapon usable once per day based on the dragon variety (see below). The breath weapon deals 1d6 hit points of damage per racial HD possessed by the half-dragon (Reflex half; DC 10 + 1/2 creature's racial HD + creature's Con modifier)." How does this work on a character with no racial hit dice and a handful of class levels? Do they just not get a breath weapon at all?
Which he replied,
Doesn't at all. A human half-dragon doesn't have a breath weapon.
