When I asked my chemical vendor, he said it will use up more chemicals, replenish more will solve the issue.
That's probably the most accurate answer.
How bad the problem is depends on the volume of your developer and how much cross-processed film you run through it. The other chemicals are not as critical; don't worry about those.
There's a big difference between using one liter of developer until you've reached its capacity (e.g. 12-18 rolls); if you throw in the occasional roll of E6 film, you'll get sooner to the point where the results start to drift. If you're running a minilab with a few liters of developer and automatic replenishment, the odd roll of E6 that goes through the machine won't make a difference. If a significant amount of cross-processed film is run through the developer, there may be problems; if it's very little (let's say up to 5% or so), I don't think you'll run into any issues.
In the above, I'm talking about running E6 film through C41 developer. The other way around (C41 film in E6 chemistry) I don't expect to be very problematic, but it's also a lot less common, so I assume you're doing E6 film in C41 chemistry. ECN2 film in C41 chemistry will not likely be any problem at all.
The main consideration AFAIK in running E6 film through C41 developer is the halide (bromide, iodide) that leaches from the film emulsion into the developer. This will affect the activity of the developer and can also have subtle effects on aspects like acutance, color balance etc. The main issue is that E6 will overall leach out a lot more halides than C41 film, simply because it builds a lot more density, so there's much more silver halides in there to begin with. I also suspect that the ratio between the different halides (notably bromide and iodide) will be different for E6. Iodide in particular will have a profound effect on the developer, which I believe is the main cause for trouble if too much E6 is run through C41 chemistry. Increased replenishment will get rid of the excess iodide. So it's not so much a case of the developer 'wearing out', but stuff being added to it by the film itself, that causes potential problems.
