How long does a lip piercing that's been in for 10 months take to close up if its been out for 2 days?

How long a piercing takes to close up depends on the gauge of the piercing but mostly it depends on the individual. All I can give you is anecdotal evidence; I removed my 14g lip piercing which was 18 months old and was unable to re-open the hole without re-piercing just two days later.

If you want to keep the piercing open but need it to go unnoticed (say, for a job interview or a particular event) consider using a piercing retainer, these are clear plastic bars which are almost invisible when worn.