Postdoc supervisor's impact when applying for a follow-up job
If you need to list references, you will almost certainly be expected to list your current boss—in this case, your advisor—as a potential reference. At the very least, if you do not, there is a non-insignificant chance that at some point in the interview process you'll be asked at will ask why he wasn't listed. You should have as good of an answer as possible ready.

Note that stating "I'm ready to move on, but he wanted me to stay on" may raise flags for a potential employer, as they could wonder whether this reflects a lack of loyalty on your part.