Ah...furries. I've had a few terrifying experiences with them in real life (no, it's not up for discussion).
Still, I'm going to strive to do better: I'm going to ignore the usual topics about the porn, the alleged persecution complex, the other usual general bad vibes of the Internet, and do some speculative scientific reasoning instead.
Why? Because I'm bored.
Now...
I imagine that most results of that sort of genetic tampering (if possible) would seldom (statistically improbable is an extreme understatement) result in that idealized "furry". It would probably end up with a partially/grown or fragmented biological form rather than a humanoid with claws, fur, and a tail.
One problem (stemming from fiction/fantasy, I know) is that the fandom seems to treat furry concepts as a method of simply altering the human form. Usually, they aren't trading more "human for animal"; they're simply tacking on features they wish they had.
While that's fine for pure fiction, if one were to venture into the hypothetical realm of human-animal hybridizing (to any significant degree), I imagine the process would never produce the intended result.
Why? Because evolution and traits don't work that way; rather, evolution seems to deal in adjustments and tradeoffs in a species over a long time. Even assuming that one could eliminate the element of time through direct alteration, you're still playing with biological disparity:
A furry wolf-man would get sharp teeth, a snout, and claws while retaining a brain sufficiently large enough for sentience. It seems rather odd, almost backwards, because our species evolved away from most of those features. In fact, human teeth are actually lagging slightly behind due to the rise of civilization (they no longer self-sharpen or grow back, yet they still haven't adjusted much to the food plaque that occurs due to the strong variety in our diet).
Why? We were slowly becoming tool users, and applying more complicated intelligence. Claws could get in the way of utility, and even flint was on par, or better, than most natural claws at the time. Can you imagine trying to use a keyboard with claws?
But for kicks, lets going further into the realm of the ideal: Wings.
A winged furry's, er, wings, in order to be feasibly flight capable in our atmosphere and gravity, would require a positively massive span, powerful locomotion, and have an average body mass less than half that of a modern, fit adult human (all that and STILL retain a strong enough body structure for a large brain and spine).
Mostly though: It's the Law of Scaling working against the furry here. (an ant that is scaled up to human size would barely be able to stand).
Oh, and being from a tool-using society, this also assumes that the winged furry isn't carrying much beyond their own clothing and is in good physical shape; not quite as practical as one might think.
I dunno about you, but I would still probably want the ability to fit inside a vehicle for long distance trips.
SHORT VERSION: While it's entirely idle speculation (I'm not a biologist), I'd guess that playing "mix and match" with animal and human traits, scientifically, would be wishful thinking at best.
