tl;dr: They may have played a small role in encouraging the anti-social behavior of the dwarves, but as they were merely encouraging the dwarves' inherent nature, I doubt the Rings' effect was significant enough to matter.
For starters, I'm not sure I wholly agree that the dwarves "caused endless trouble to the elves" during the Third Age (or the Second, for that matter.) The two races clearly did not like each other, but the dwarves mostly remained underground and the elves mostly remained in their forests. There was active trading between the races during the Second Age, and the dwarves participated in the Last Alliance for a while. In the Third Age, there was definite cooperation at certain points (e.g. the gates of Moria), and both elves and dwarves participated in the Battle of the Five Armies. However, by and large, the dwarves began to shut themselves away from the rest of the world near the end of the Second Age and didn't really emerge after that, even during the War of the Ring, unless they were forced to surface by dragons or balrogs.
As far as the Rings of Power themselves, we don't know a whole lot about the ones the dwarves had. The short list is:
- In The Silmarillion we learn that the rings had only a mild influence on the dwarves. Sauron could encourage them to anger and greed via their rings, but they were otherwise unaffected (naturally resistant).
- By the time of The Hobbit, the dwarves had lost all of their rings; dragons ate 4 of them and Sauron recovered 3.
- In Lord of the Rings Gandalf repeats a rumor that the rings formed the start of the Dwarf Lord's hordes.
We don't know how long the dwarves held on to their rings, nor how many of them were "in play" during the third age. We do know that at least one of them was in the hands of it's bearer until around III 2850 (the "Ring of Durin/Ring of Thror" that is mentioned at Elrond's council, actually held by Thorin's father until Sauron captured him) but that line of dwarves spent a large part of the third age fending of monster attacks and looking for a home, and probably not causing problems for the elves. I don't think we even know the names of the dwarves that got the other seven rings (presumably they would all have been descendants of the seven Fathers of the Dwarves but otherwise, who knows.)
(For some reason, I always have always assumed that the dwarves didn't even wear their rings but rather kept them locked away with their rest of their treasures. I cannot find any reference for that last bit, though, so I may have made it up.)
Given that the dwarf/elf hatred dates back to the First Age, the fact that the rings only real effect was to magnify their already overwhelming greed, and that in the grand scheme of things, the dwarves probably caused more trouble for the elves by just avoiding them than anything they actively did, I don't think the Rings of Power had any significant impact on their behavior while they were active.