There's no general answer to that. It depends on where your ELISA antibodies bind. Does the IL-22 binding site block any of the ELISA antibodies' epitopes? If yes, IL-22 will obscure the signal. If not, it won't.
If you use monoclonal antibodies in your ELISA, then this will usually be clear-cut. Either IL-22 blocks your signal completely or it doesn't. You have a chance of finding this out by calling the manufacturer of your ELISA kit and asking them what the epitopes are. They may or may not know. Then, you might line that up against the sequence and 3D structure of your IL-22-IL22BP complex to make a prediction. Or, probably easier, just buy yourself some recombinant IL22, add a large excess to occupy all your available IL22BP standard and see what happens.
If your ELISA is polyclonal, you'll have all sorts of different antibodies binding all sorts of different epitopes. Some of them will be blocked by IL22, and others won't. So in that case, you can expect a partial loss of signal following a probably highly non-linear dependency on your IL22 vs IL22BP concentrations. It might be mathematically tractable to get quantitative results in this case, or might not be. Furthermore, the cocktail of antibodies in your polyclonal preparation might change with each production lot -- each time you vaccinate a goat, it might make an entirely different set of antibodies in different quantities. Again, try and see what happens, but you might need to accept being out of luck if you have a polyclonal assay.
You are absolutely right, this is in fact the reason for my question. I should have specified that I am wondering if someone has any experience with a commercially available ELISA to quantify IL-22BP (for instance the kit from R&D Systems). Epitopes are generally not disclosed for such kits and I'm reluctant to use one, since I don't know if bound IL-22 will interfere with antibody binding.