From Tails, using Thunderbird with it's default settings? No.
If Tor's connection dropped or Tor crashed you'd just lose connectivity entirely.
Any programs correctly configured to use Tor's SOCKSPort, like Tor Browser or Thunderbird with TorBirdy installed, would just throw errors about being unable to connect either to or through the proxy. They would not "fail open" (I.E. circumvent Tor)
I guess it's worth stating, for factual accuracy, that software can and does go wrong, and it is possible that an unlikely confluence of unforseen software behaviour happening in both the kernel and the program at the same time could lead to this...but it is not a very realistic worry.