If you were fairly new to a position, would you ask for time off after your supervisor requested that you work over the holidays?
Probably not, and definitely not if I'd already agreed to the request.
For this example, let's say your friends asked that you accompany them on a trip that you would really enjoy.
Your supervisor only recently asked that you work, and you were already contemplating going on the trip, but had not made firm plans or requested time off.
That was your chance to bring it up to your supervisor and request the time off, so the schedule could be covered. Did you agree to work during this time?
Your supervisor asked you to work days that are not part of your normal schedule over the holidays, however some of the days you want off are days you do typically work.
Not unusual for holiday-season work, when people have to cover for others who are on holiday. I don't see that fact that some of these days are not your regular days as relevant, if you already agreed to work them.
Would you still go if your supervisor told you it was a problem, but did not deny your request?
Only if I had plans to move on to another job when I got back.
How would you feel if your request was denied and you had already made vacation plans?
I guess I would feel very foolish for having made formal travel arrangements before making sure I had approved time off. I would also understand why my supervisor would deny my request, if I'd already agreed to work and knew that there was no one else to cover my days. (Though, all this is moot, because, in this situation, I wouldn't be asking for time off at the eleventh hour unless it was an emergency.)
All in all, this seems a best an exercise in futility, as it would be very hard for your supervisor to grant you the time off, and at worst an extremely good way to burn bridges with this job.
It's unfortunate that the timing didn't work out, but if you agreed to work those days, you're stuck, if you want to keep a good reputation at this job.
As a manager, for an employee in your situation to even ask me this, knowing the circumstances of there not being cover and having previously agreed to work the shifts, it would definitely lower my opinion of them. So there is "something to lose" even by asking, IMO. Sorry.