The problem with their plan is that yes, 16 Republicans abstaining would kill him as Speaker, but just 1 Democrat voting for him would kill the abstentions. This only gives Obama even more leverage as he can pull the negotiations further to the left by promising #SaveBoehner matching the Republican abstentions with just a handful of Democratic votes for Boehner.
Good point, the Tea Party trying to do this would probably only hurt themselves by giving the Dems leverage. It would be terrible optics for Boehner to have to rely on Democrats to win, though.