I'll do my best to answer here. I don't know what field you're in but my answer is going to be based on my experience doing social sciences research.

The answer to your question depends on why you think the residuals are correlated and what you are going to be using the model for.

The residuals of two or more indicators can be correlated for a variety of reasons - examples could include: 1) the indicators share some common factor that is unexplained by the latent factor they are loading on, 2) when using a psychometric survey, correlated residuals may be indicative of redundant content (in which case, if you have significance cause to, you may remove one of the two indicators from your model), or 3) may be indicative of a systematic response bias. In the last case, I would avoid correlating the residuals and would instead model a new latent factor representing the source of response bias (e.g., common response bias).

The second issue pertains to what your model will be used for. If you plan on simply reporting the fit statistics in your paper, it would be okay to report the fit of your model if you believe that the residuals are correlated due to redundancy of the content of your indicators. If you believe that the residuals are correlated for other reasons, correlating the residuals might result in you artificially over-fitting your model. If you are going to be using factor scores based on your model for further analyses, I would say "yes" keep the residuals correlated in the case of reason 2 (mentioned above), but "no" in the case of reasons 1 and 3. In these latter situations, a better course of action may be to create a new latent factor (provided you have a very good reason for it!!)

Essentially, you want to be sure that the model you are reporting (and potentially using in subsequent analyses) is an accurate depiction of your data and that you are not modeling extraneous factors such as systematic response bias. In order to determine this, it is necessary to revisit the indicators in your model and think critically about why these residuals are likely correlated.

Hope that helps.

Thanks for your answer! The correlated residuals are a big trouble because the "why happens that?" is hard to answer, but the topics "method effects" in psychometrics are a great approach. Thanks again.

Dear Sergio, I suggest you to take independent samples of each factor treatment, and of the interactions. In such a way you assure incorrelation between them (and between the residuals) from the hypothesis.

Dear Sergio, I think that correlated residuals is a consequence of multidimensional Normal hypothesis for the data analysis. A supposed statistical model can have these effects.

If there is a legitimate theoretical rationale for correlated residuals in the model than there is nothing wrong with specifying them within a bifactor model. The only thing that is precluded in a bifactor model are item cross-loading on multiple group factors.