Is there a danger if the tyre or the inner tube suddenly burst?
If a tyre or its tube bursts while cornering there's a strong chance of it losing grip. Losing grip on the front while cornering is quite likely to result in a crash, less so on the back, but if it's because the tyre is suddenly flat you may not be able to recover.
On the straight it's less likely to be serious. I've had a front blowout on the straight and I wasn't close to coming off. But I was going slowly to stop under the next light and investigate the dodgy noise (which turned to to have been the tube bulging through a slit in the sidewall and rubbing on the fork). A rear blowout on the straight is probably the most common, and the safest form of blowout. I've seen a few happen and they've never resulted in a crash.