Can I sue a bar for kicking me out if I then subsequently got a DUI?

You can attempt to sue anyone, however courts frown upon filing frivolous lawsuits and this is what this appears to be. Drunk Driving is considered an offense of personal responsibility, meaning that even if the bar served you until you passed out, it was your responsibility not to get behind the wheel drunk. You can try to sue based upon neglect of a duty to stop serving you after a certain point, but I'm not certain this will suffice either.