The question "given the inverse/a certain factorization of $A$, can I compute cheaply one of $A+B$?" is one that gets asked a lot here, in slightly different forms: for instance when $A$ is symmetric, or $B$ diagonal (and I have seen it pop up several times on MO as well). The answer is always "no, unless $B$ has small rank (and then you can use SMW or QR/Cholesky updates) or small norm (and then you can use a perturbation approach and/or use $A$ as a preconditioner in an iterative algorithm).
I think it would be useful to have a canonical question in which we explain this in full generality, so that we can refer people to it instead of re-explaining the same thing continuously. What is your opinion? I volunteer to write one in the next days, if you wish.