Um... yes?
I don't think either team wants to do the Celtics any favors. Especially after Danny ripped them both off in those deals pretty badly.
The difference is Dallas could tank if they wanted to and not worry about it. Brooklyn has no choice.
I don't agree with the bold part here.
Danny didn't rip any team off here. The teams new what they were buying in to, they knew the risks, they took the deals anyway. They gambled and they lost.
Danny took a gamble all the same. Brooklyn may have won two titles and our pick this year cold be worth nothing. Rondo could have elevated his game and become an all star again, and Dallas could have spend the next two seasons competing. Crowder and Jerebko could have sucked badly, failed to earn playing time, and been let go during free agency.
All three teams made their bets, and Boston won the jackpot. If Dallas and Brooklyn are grump about losing so big, then they should be grumpy at themselves for taking such huge gambles.

Regardless of what you want to call it, Danny won those deals by a pretty fair margin, and my point really was that Dallas and Brooklyn won't want to help benefit the Celtics' future with those picks. So to answer the OP's rather obvious question, yes, the Mavs and the Nets will purposely try to win as many games as they can this season.
The picks are gone and no decisions either team makes will have anything to do with them. You seem to imply they might try to win somehow out of spite and that is ridiculous. Each team is going to do what they need to get better in the future for the franchise, not a lost deal.
So wait, you're saying that Dallas and ESPECIALLY Brooklyn wouldn't be just a little mad that Danny gave them faulty players (EDIT: Perhaps "faulty" aint the right word. I can't put my finger on the right word but my point is the guys they acquired DID NOT HELP THEM) that have hurt their future? Maybe not Dallas as much because the picks not unprotected, and it only hurts them in the short term, and they didn't give up a huge amount for Rondo, but Brooklyn mortgaged their entire future until 2018 for KG and Pierce and it backfired horribly. I agree that both teams have no one to blame but themselves for making those deals, but at the same time, I think they might just be a little angry about how much Danny and the Celtics won out, and now he has their picks that could very well be in the lottery.
Now, when both deals were made, I don't think anyone really thought Danny was going to make the big score that he did on these deals, but they turned out better than I bet even Danny was expecting it to. I guess the reason why I view it as a rip off is because KG/Pierce and Rondo turned out to not be the final pieces that Brooklyn and Dallas expected them to be when they were aiming for a title. In fact, the deals turned out to be the opposite of what either Brooklyn or Dallas wanted. Now Brooklyn's failure IMO was not on KG and Pierce. They did what they could for that team. I blame that Brooklyn put too much faith in Williams' leadership and Lopez' health. Why it's a rip off is because they gave away 3 unprotected picks and a pick swap for that. With Dallas, I think Rondo ruined their season. They relinquished their depth to get him because they thought he was an upgrade at point guard and he turned out to be the worst thing for that team. I think Brooklyn feels ripped off because of how much they gave Danny in the deal. I think Dallas feels ripped off because Rondo not only failed to be the final piece, but also his conduct ruined the team.
I feel like what I said has been misunderstood. All I was doing was answering the question. Are the Nets and Mavs purposely not tanking due to the picks owed to the Celtics, and my answer was without a doubt, yes, the Nets and Mavericks are trying to win this year. I feel like I have said that several times now. Now the reason why I said I don't think the Nets or Mavs want to do the Celtics any favors is because the Celtics own their picks. I also feel like I've said that quite a few times now. Is that not clear?
The Mavericks could hypothetically throw their season away and not worry about the Celtics taking their pick, but from the looks of things, it looks like that's not going to happen. The Nets on the other hand really have no choice in the matter because their pick belongs to the Celtics for the next three years whether they like it or not. I don't think either team has the mindset, "Hey, remember when we gave Danny Ainge our picks for guys who we thought were going to help us make a playoff run and in the end they didn't? You know what we should do? We should suck and hence give them a lottery pick!" Especially Brooklyn again not just because they're stuck but because its a division rival.
Are the Nets and Mavs going to win primarily to spite the Celtics? No, and I never wanted to make it sound like I was saying that, but what I was saying was now that they are in the situations that they are in, they are going to try to win at all costs, but one of the reasons why they won't tank is because the Celtics own their pick and they don't want the Celtics to benefit that much from the deals they made with them.