(Warning: Contains spoilers)
So, when we last see the Deadly Six, Sonic defeats them in battle. Zazz, Zomom, Master Zik, and Zeena explode in a puff of smoke while Zor and Zavok are sent falling into lava. This would suggest that they are dead, and has me convinced of it, but a few things might convince others that they survived.
First, when Sonic defeats Zazz, he says "The boss is gonna beat me!" and explodes a second later. His line suggests he'll still be around to be beaten, but maybe he didn't know he was mortally wounded at the time. Compare Crystal King from Paper Mario, although he is equally debatable.
When Sonic ascends Zeena's battle arena in Lava Mountain, she says she should feed Sonic to Zomom, when Sonic already killed Zomom by this point. It could be argued that Zeena, as well as Zavok and Zor, were unaware of Zazz, Zomom, and Zik's fate, as they were simply out doing their own thing at the time and, not witnessing their deaths, either never tried to contact them, or did and simply thought they were slacking off.
After Eggman's defeat, he makes plans to get another conch and control the Deadly Six again, making them more powerful. All of them have been presumably killed by this point. But maybe Eggman didn't know of their fate, because after he faked his death with the lava fall, he was building his giant mech, so he would have been unaware of the battles that were going on. If they had survived, how would Eggman have known it, because they don't show their faces again after their final defeats. Furthermore, he says he'll find another conch, when he implied that he wouldn't be able to find another one easily, and may have not been in his right mind after that battle. Or he could have even planned a revival with Chaos Emeralds. (Or does that require a body within a certain amount of time after death?)
Explanations that they might have survived, and counterarguments: For Zazz, Zomom, Zik, and Zeena, it is possible they teleported away. However, if they had such a power they likely would have used it the previous two times Sonic beat them, but instead they jump away. The only time we see such an explosion is their final defeat scenes at Lava Mountain. Furthermore, this is how Super Mario Galaxy bosses go (although some of those turn up alive again, remember they're in the Koopa Troop, who are established to be able to revive teammates). And also, before they vanish, they lay still on the ground. If they were supposed to survive, they'd have jumped away for the third time.
For Zor and Zavok, it's possible that Zeti can survive lava. However, Zor's line about "longing for death's cold embrace" right as he falls into the lava, suggests otherwise. Also, it can be debated whether Zavok fell into lava at all, but as the lava seemed directly at the bottom of the shaft, I think he did. Again, the only two times a Zeti takes a lava bath is the final fights with Zor and Zavok, and they never reappear after that.
While Zazz, Zeena, and Eggman's lines leave open a slight possibility for their survival, there is nothing that confirms or strongly hints that they're not dead. Contrast Eggman. At the end of the game, he falls over a cliff, which had me thinking he was dead until the postcredits scene shows that he survived. But it was only his survival that was confirmed, the Deadly Six are never seen again in the game after their respective defeats in Lava Mountain. So I definitely think they are dead, but to the other fans, their status is about equal to Black Doom - most likely dead, but not 100% confirmed. (Also compare Queen Chrysalis in a certain other show, she's almost as dark as the Deadly Six I might add, also King Sombra had a similar debate before the show's creators confirmed it)