2 good answers to this already (one here, one in world building).
To put some very rough estimates to it, CH4 or Methane has increased significantly over the last 250 years from about 0.72 parts per billion (ppb) to about 1.83 ppb. Source. This accounts for about 20% of the current man made warming, which might be estimated at 2 degrees C by 2100 (very rough estimate). If CH4 is .4 degrees C of that warming by 2100, that's not nothing, but it's still a significantly smaller problem than CO2.
What's more, CH4's half life in the atmosphere is shorter than CO2's, 12.4 years vs over 100 (same source).
If all the Tundra on Earth was to thaw at once (very unlikely), we'd see a big spike in CH4 and perhaps an uptick in yearly warming, but within 12 years, half of that extra CH4 might be out of the atmosphere. If the Tundra warms and thaws more gradually, which is more likely, we'd see an increase in CH4, but over time, the increase would level off. It's not like CO2 which continues to build because the earth's systems can't take out out of the atmosphere nearly as fast as we put it in.
So in and of itself, CH4 from melting Tundra isn't a big concern and turning the sky brown or poisoning birds, as others have said, that's very unlikely. Perhaps it will turn our sunrises and sun sets slightly redder. (perhaps), I'm not even sure on that point.
The CO2, as Gordan points out, is the bigger problem and precisely how world breaking a problem that is, or how soon it is till things get obviously problematic because of CO2 remains to be seen. It's enormously difficult to predict with any accuracy, but it's possible that CO2 is a real problem that deserves our concern and effort and I say that without trying to sound like a fear monger. It's simply the truth.
Thawing Tundra will give a little push to already existing climate change, that's commonly called a feedback mechanism and such mechanisms are already included in the climate models by the IPCC and others. The thawing tundra is part of the by 2100 IPPC estimates. It's not something to worry about on top of what's already being estimated.
Now, on "extinction level events" or extinction events. In a very real sense, the increase in human population and our taking over more and more of the earth is already an extinction event and has been for the last 2 centuries or so. Extinction events are relatively common, at least, geologically speaking, even the formation and recession of an ice age can cause a significant rise in extinction and what could be called a minor extinction event.
Now if you're talking Major Extinction Event, the last once of those was 66 million years ago when that Mt. Everest sized asteroid crashed into the earth and was the primary cause in killing off the dinosaurs.
So, part of the answer to your question is a problem of definition. I think the thawing Tundra, by itself isn't much to worry about, (unless you have a house built on frozen land that turns to slush under your foundation) - and yes, that happens fairly often) . . . but on a planetary scale, The thawing tundra in addition to the CO2 and warming planet is maybe 10% or 20% max added to an already existing problem, so the larger worry is the combination, not the Tundra in and of itself.
There are some methods already in use to help keep Tundra frozen, like thermosiphons
Brown skies and toxic air are statistically quite unlikely and certainly wouldn't be driven just by thawing tundra. It would take a lot more than just that. The ocean current stopping might cause such a problem, but trying to work a prediction around that beyond saying it's unlikely, is very difficult, even with everything we've learned so far.