In the first episode of Orange Is the New Black they mention that Piper has a 15 month sentence. In the last episode of Season 1 it's Christmas. Are there any hints or comments I might have missed that indicate how much time has passed?
I believe the answer is between 3-4 months. I hope that's accurate enough for you. This is based on :
Halfway through season 2 (I can't remember which episode) Piper mentions she's served 5 months of her sentence. At that point in the series at least 4 weeks have passed based on events that have transpired.
Season 1 ends at Christmas, and Thanksgiving is episode 9.
Season 1 episode 8 has Pipers friend giving birth, at the beginning of the series she is heavily pregnant, although it's not mentioned how far along this pregnancy is, friends with experience tell me 7 months would be a safe estimate.
Totting all that up gives an approx answer of 3-4 months passing in series 1.
I wouldn't be surprised if the timeframe was kept deliberately ambiguous though to allow as many future series as is possible/profitable.