In IHC, designing the experiment correctly is very critical. In the ideal situation, you will label unique proteins or antigens on the tissue, without interfering with the other labeled markers, so choosing a distinct type of marker – either nuclear, cytoplasmic, or membrane marker and not overlapping them. It is recommended to use the DAB chromogen first, then the red chromogen.
If both antibodies are against the same site (e.g. both recognize a target in the plasma membrane), then you might observe that the dyes affect each other. The outcome is unpredictable but what might happen is that the strongest chromogen will dominate. However, much depends also on the specificity of the first antibodies. If the antibodies are highly specific and do not overlap at the epitope binding site, then a clear and crisp chromogen staining could be obtained.