I would try some variation of the following:
Make the logo black in a separate comp.
Add the comp three times on top of your wood image.
The bottom most layer would have Roughen Edges followed by a fast blur – Animate Roughen edges and fast blur to simulate the darkening of the wood as it is being burned. (should start lighter and thinner and bloom to fully charred)
The next layer would be the same but would not bloom out past the edges of the logo – use this layer as a track matte for a red/orange noise layer to simulate the red embers as the wood chars.
The top most layer would be combined with your partical simulator of choice to get some smoke comming off of the logo – more at first then just wispy trails after full char.
Good luck!