There was a little bit of surprise when the 5 PM deadline for teams to make qualifying offers to their pending free agents came and went this evening without the Pirates making an offer to AJ Burnett, so before going into the mechanics of the Pirates' decision here and what they might mean, I figured a quick refresher course on the "qualified offer" system is probably in line, especially since it's only in its second year and this is the first time the Pirates have been in position to take part in this process.
When the new CBA was ratified, the old A/B free agent ranking and compensation system was scrapped. The way the old system, that people are mostly still familiar with, worked was that if a team had a Type A or Type B free agent, they had the option to offer arbitration to that player. The player could either accept arbitration, at which point the two sides would try to work out a deal before arbitration in the spring, or they could reject and hit the free agent market, with a draft pick price tag. Type A free agents cost teams first round picks (depending on record), while Type B free agents rewarded their old teams with sandwich picks. The ranking system was arcane and skewed and exploitable, though (in some cases, teams would keep or trade for Type B free agents specifically for the compensation pick after the season), and plus nobody likes going to arbitration, so the whole thing was scrapped for the new "qualifying offer" system. Now, the top free agents at each position are eligible for a "qualifying offer" from their original team that is a set amount of money ($14.1 million this year), provided that they didn't join the team mid-season (hence, Marlon Byrd and Justin Morneau were not eligible for qualifying offers from the Pirates, though they would have been had they stayed with the Mets and Twins). They can either accept that offer, negotiate a deal with their original team, or become a free agent and give their original team compensation when signing with a new club.
In short, what the Pirates not extending this offer to Burnett means is this: if he signs with another team, they have nothing to show for it. The Pirates can still sign Burnett, they can sign him for $1 million or $13 million or $20 million; this decision does not restrict their ability to re-sign him. If you operate under the assumption that Burnett has been truthful when he's said that he has no intentions of pitching for another club in 2014 and he'll either pitch for the Pirates or not pitch at all, no qualifying offer was necessary. Burnett's been saying this non-stop since April, and he re-iterated it to Colin Dunlap when the hubub around his availability for Game 5 of the NLDS arose, so I don't see any reason not to take him at his word. The Pirates have been able to talk and negotiate with Burnett since the season ended, so presumably they have an even better feel for this than you or I do.
There are a few ways to read into this, should you want. The easy reaction is just to say that this is that the Pirates being cheap and that they want the negotiations to start somewhere below $14.1 million. I don't know that that's the case, but it is pretty tempting to argue that $14.1 million is a fair a very fair price for a pitcher like AJ Burnett and that the Pirates should've taken out the insurance policy in the form of the qualifying offer just in case. I do think that it's worth considering the alternative option, though, which is that the Pirates and Burnett have already agreed in principal to a one-year deal that is worth less than $14.1 million and that all Burnett is doing now is weighing the merits of retiring vs. returning.
Either way, I don't expect this decision to drag out. The Pirates will need an answer from Burnett relatively soon so that they can map out a plan for the off-season and have a free agent budget in mind, and there really haven't been any indications from Burnett that he's going to keep them waiting. Until we know how this all actually shakes up, speculating on the decision to not make him an offer today seems a little premature.