No, and it is impossible in principle. MLIR doesn’t have a fixed set of operations, so it wouldn’t know how to execute the operations from your dialect.
If you set up a pipeline that can lower your mix of dialects to the LLVM dialect, you can clone the module after each pass, use the pipeline to convert the cloned module to the LLVM dialect and use the ExecutionEngine to run it. In many cases, you can target a higher-level dialect for which the lowering to the LLVM dialect already exist so you won’t need to test it yourself (it’s unlikely, though not impossible, that the in-tree lowerings are buggy).
Alternatively, you can implement an interpreter for your mix of dialects.