I installed a bathroom fan in the ceiling. My bathroom is in the room right next to the laundry room. The washing machine has the standard drain pipe vent of a pvc pipe venting to the outside through the roof. To avoid cutting another hole in the roof I connected the bathroom fan vent to the washing machine drain vent in the attic so it can vent outside. The only problem I can foresee with this is maybe some rain could possibly get down into the bathroom fan, but that is very unlikely the way I have it set up. Any other issues or problems I can't see?
You have two big issues here.
First is that you have a serious risk of venting sewer gasses into your home. This is why drain vents open above the roof line or use one-way air admittance valves. This is likely to be unpleasant and possibly unhealthy.
Second is that a drain vent is typically too small and too moist to properly vent a bathroom fan. Your fan is probably designed to use a 4" duct, and instead it's going through a 1.5" diameter pipe. And if any lint/dust gets pushed out by your fan, it risks accumulating on the sides of the vent pipe, potentially blocking the pipe and slowing both your drainage and ventilation.
Beyond those functional problems, you'll also have a pretty visible issue that anyone inspecting your attic will see. You should do this the right way: add a properly-sized duct line and a roof vent for ventilation, and re-seal your drain vent pipe.
Not enough detail to be sure, but I doubt that the drain vent has a large enough cross-section to allow enough airflow. The vent pipes in my house are 2"; the ventilator conduits are 6" or 8". You may find that your fan is spinning and making a lot of noise but not moving much air...
Also, your statement that it's "very unlikely" that rainwater will get into your fan does not fill me with confidence. The downside, if rain does get in, isn't just that the fan might stop working - it very well might catch on fire*. So make absolutely certain that rain won't get in, OK?
*My sister's bathroom ceiling fan caught fire a couple of months ago - albeit due to advanced age, not rain damage - and came very close to burning down the house. So this is a topic that concerns me.
The answer is no way! Sewer gas is explosive and could settle back into your exhaust fan which has an electric motor that is not rated for explosive gases. You're now not at code for the sewer or the fan. If a fire or explosion were to occur, good luck on your insurance claim and worse if someone was to be injured. Vent the fan through the roof as required and fix your sewer vent system.