It sounds like you are working the Danish oil too much, whichever dye you use. You have, however, found one the challenges inherent in applying a wipe on/wipe off finish over wood that's been coloured with dye or stain. Oil based dyes are usually formulated with white spirit and/or naptha, which also happen to be solvents in Danish oil, so the white spirit and/or naptha in the oil is dissolving the dye. The solvent for spirit dyes are usually alcohol, although cellulose thinners and other solvents are also compatible, but again the Danish oil can have a tendency to lift the colour because there's often a bit of methanol in it. You would get some lifting of dye colour even if you used water based dyes.
In reality, what you need to do is to get the first coat of Danish oil on quickly and thinly, and then not mess around with it, and especially avoid rubbing and burnishing the applied finish. To do this apply it in thin, even strokes with a high quality brush and avoid going over and over the same spot because all this does is pick up dye that's been re-dissolved and move it around leading to patchiness. Once you've got the first coat or two on like this you should be able to apply subsequent coats in the normal manner and wipe off as described on the can.
Danish oil, as is the case with many of the finishes sold as 'oils' e.g., teak oil, is in essence a sort of varnish 'lite' because the main constituents are usually naptha or white spirit (usually a significant percentage of the formulation, possibly 30- 50%), something like boiled linseed oil or perhaps pure tung oil, and a resin that dries to form the film: there are often additional constituents such as a tiny bit of methanol (as already mentioned) and heavy metal driers. So, if you can get the first coat or two on without re-dissolving and moving the underlying dye about you should have more success.
In reality, if I want one of those 'close to the wood' appearances which is what you get with Danish oil, I simply grab a can of oil based varnish, put 30- 40% white spirit in it and apply two to four coats of that. It's just as easy to apply and a hell of a lot tougher. Slainte.