Nylon is similar to animal fibre in terms of dying. You can use dyes for polyester on it, but I think acid dyes, like you would use on wool would be better. 'All purpose' dyes like the dylon machine one will not work.
I think that food colouring and kool aid also work, again like they do on wool. I think you need vinegar to set the dye. I've not tried it myself, but I know some wool-dyers, and we had a conversation about sock yarn with nylon in it for strength, which takes the dye the same as wool.
Whatever you use, you need heat to set the dye, so I suppose there might be a risk of damaging the dress there.