Turtles Forever/WMG

The movie inadvertantly establishes that the "red sky" seasons of the 1987 cartoon are in a branched-out dimension all its own.
The atmosphere and art style of the show was a lot darker in the "red sky" seasons; and from what little we see of the 1987 dimension in Turtles Forever, it's still got a blue sky, a VERY lighthearted atmosphere, and the original art style in place. Not to mention the fact that the last time we see the Technodrome in the "red sky" seasons, it's an abandoned wreck in Dimension X, and, more importantly, it's still in its original form...it has none of Ch'rell's modifications from Turtles Forever.

Ergo, the "red sky" seasons exist in a branched-off dimension that's similar, yet completely separate, from the rest of the 1987 dimension.

However, the "blue sky" universe--the 1987 dimension seen in Turtles Forever--still continued on separately. The last episode of the original show's seventh season (the last "blue sky" episode) establishes that the Technodrome gets sucked back into Dimension X, stranding Shredder, Krang, et al. on Earth. At some point, Shredder and Krang somehow retrieve the Technodrome and wreak havoc upon New York once again, and the Turtles intervene, thus leading to the events that get them stuck in the 2003 universe. The only forseeable problem is, Shredder and Krang return from that adventure with a drastically upgraded Technodrome, care of Ch'rell's modifications.
 * Alternately, Turtles Forever happened in one of the earlier seasons of the 1987 show. There was no mention of those turtles being at the end of their adventures, and when we saw the city streets, it was still the '80s. The Technodrome snapped back to its original state and the show continued on the path we saw in elementary school.
 * It'd have to be one hell of a Snap Back, though. It was shown that Technodrome 2.0 could travel to different dimensions without changing in form. Not to mention that it would take quite a bit of manual labor to downgrade it, and even Krang wouldn't be stupid enough to downgrade a flying Technodrome with a giant Death Ray. On another note, if you recall, the special didn't say anything about any of the Turtles seeing an end to their adventures--for instance, Karai insists that Ch'rell will return, and Hun is still a giant mutant turtle, so the 2K3 Turtles will have plenty to do upon their arrival home (even if the viewing audience never gets to see it). The same could be said for the 1987 Turtles, who now have to deal with Technodrome 2.0.
 * Don't worry about the Upgraded Technodrome, Bebop and Rocksteady will continue to wreak the 1987 Shredder and Krang's plans, no matter how sophisticated the new technology is, those two will do something stupid. And the 1987 Turtles will eventually find a way to fight the new ninja bots.
 * Most likely, sometime after the last episode of the 1987 show and before Turtles Forever, Shredder and Krang were able to have the Technodrome rebuilt with help from Krang's followers and slaves. They are now back on Earth, but the Turtles stopped them from bringing Krang's army to Earth. So now they're back to the way things were in the old days until they ended up in the 2003 universe.

Ch'rell has ascended to a higher level.
The beam didn't just destroy Ch'rell, it crossed him over into a new level of existence. He becomes a cosmic being, roaming across time and dimensions. Because of this he is freed from the evilness and hatred in his soul, and realize the errors of his ways. Or his hatred for the Turtles is stronger than ever, and it would only be a matter of time before he'll have his revenge again.
 * An interesting theory...how are we to know what would happen if a being gets killed in a dimension not its own?
 * Alternatively, maybe Ch'rell had a transdimensional fail-safe built into his robot suit in case of an extreme emergency. 1987 Don proves that a tote-bag sized transdimensional transporter is possible, and we never see the Utrom itself get vaporized, so perhaps Ch'rell got beamed to another dimension as a means of escape.
 * Alternatively, Ch'rell might've been sent to the universe of the 2012 TMNT series being made for Nickelodeon, and the 2012 TMNT will have to team up with the 1984, 1987 & 2003 TMNT to stop him.
 * This troper thought the same thing. However, characters from the other continuities won't make an appearance since the 2012 cartoon will likely be self-contained from the multiverse. Ch'rell will exist as a spirit taking the form of his Shredder armor, and influence the Shredder of the new cartoon. It won't be revealed that its Ch'rell, but it'll be obvious.

With exception of the Mirage Turtles, which sticks to the their first comic history like glue, the Turtles are from different universes from their TV Shows.
For I.E.: The 2000s Turtles are from universe in which Cyber-Shredder never happen--but they did get trapped in the Future--mostly like dealing with an alien version of Shredder and Cody's evil uncle. They return to their own time without Cody's robot following them. The 1980s Turtles are from a universe in which Technodrome just got it's tank wheels removed and stuck in a small Forrest in New York. Likely, it was a easy repair job for Krang, Shredder, Bebop and Rocksteady-allowing the Technodrome to move into New York to do stuff-leading to Shredder's ooze robbery and the explosion that lead to the crossover.
 * Except their Cyber Space technology is referenced in the movie suggesting Cyber Shredder did happen.

The 2003 series is set at the same time as the 80s one.
No one made mention to the fact they were in the future or past. So either the 2003 one is Schizo-Tech or the older one just lacks technology in exchange for goofy retroness. I support the latter. It seems more in-series for the 80s one to just be so goofy that it has an 80s charm (which is pretty goofy) and lacks modern day technology.
 * The Technodrome computer did make note of a "temporal anomaly", which could indicate that some form of time travel came into play. Or maybe the '87-verse and the '03-verse have their time streams synched in some kind of significant way, even though both series technically should (and do) take place decades apart.
 * They're synched by the Turtles. After all, they're the keystones of the multiverse, right? So it makes sense the TMNT universes would be synched by the origin of the Turtles within each.

The 80's Turtles are the end result of a Seasonal Rot that never ended in their universe.
In the first couple of seasons they did have distinct personalities, take things seriously, and were less goofy, while Shredder and Krang were at least mildly threatening. But if you keep watching into seasons 3 and 4 up until before the Red Sky seasons, they approach the goofiness in Turtles Forever. The ones we see in Turtles Forever must be from a hypothetical season 22 (given that the special came out in 2009), in which the series has long since Jumped the Shark and everyone's been Flanderized.

Krang knew and expected that Bebop and Rocksteady would manage to mess up Ch'rell's plans
Even as the multiverse is being pulled apart at the seams, he doesn't seem too worried about it. "May he find you as useful as we have" was less a snide remark at their incompetence and more Krang recognizing a pattern in what happens to anyone Bebop and Rocksteady ally with.

The '87 Turtles and universe were a last-minute addition to the movie
There are certain moments where they just don't "gel" with everything else.