One common assumption is a new general manager would want his own people running the Jets, and that could spell the end of Rex Ryan. Most organizations have the general manager in charge. If we know one thing, however, it is that the Jets are not run like most organizations. A Mike Tannenbaum dismissal might end up increasing Rex Ryan's power in the organization.
On some level, there seems to be a power struggle between Ryan and Tannenbaum playing out behind the scenes. It might be that both will be shown the door. It might be that a new general manager would be charged with dealing with the head coach situation. It is also possible that Rex could outright win the struggle. The Jets might interview general managers with the stipulation Ryan keeps his job. It's how Brian Schottenheimer was retained after Rex was hired. Ryan might even be given a say in the choice. While uncommon, a situation like this is not unheard of in the NFL.
I do not have any inside information on how likely this is. I am not saying this is how I think it should go down. I wouldn't be shocked to see something like this, though, based on how the Jets have functioned.