Do Tallbird nests reappear after being burned?
It is not explicitly stated, but the Don't Starve Wiki has tallbird nests listed as non-renewable resources. I take this to mean that they do not spontaneously re-spawn if burned.
The List of Non-Renewable Resources
That said, if you wish to cause one to spawn you can have the game spawn one for you.
In-game hit ` and type:
DebugSpawn("tallbirdnest")
A tallbird nest with an egg will instantly appear in front of you. It will hatch immediately into a tallbird who will start to attack you for being so close to its nest!
To make the console disappear hit Control-L.