The tires on my bike are inflated to 5 bar above ambient. Ambient of course being one bar. If I were to take my bike into outer space, that would mean that all of a sudden the pressure is now 6 bar over ambient, or a 20% increase. That would stiffen up the ride a bit but probably not burst them.
Aircraft tires, at least the ones on the PA-28, are inflated to 1.6 (nose) to 2.0 (main) bar over ambient. Taking them into outer space makes this 2.6 and 3.0 respectively, an increase of about 62% (nose) / 50% (main) max. That might be enough to burst them. But indeed, you have to take the PA-28 into outer space, and you have to disregard temperature effects. At more realistic altitude, say 10.000", the ambient pressure is still around 0.7 bar so you're talking about a 20% increase in the nosewheel max.