Personally,I would run the risk of taking the old ones down in order to do a more thorough paint job.That is if you are replacing them.This allows you to repair any potentially bad fascia/soffet and give everything a good coat of paint ,especially areas that you could not access if the gutters/leaders were up.
Absolutely take down the old gutters. In addition to what georgieboy wrote, it will give you a chance to find any areas that could use some shoring up concerning the roofing edges behind the gutters. Squirrels and raccoons like to probe those areas and often get into your house that way.
i would take the old gutters down, paint and then install new...especially if your trim is white...then you don't need to paint the newly installed gutters which are usually white.
We dealt with this predicament last year. We had the gutter guys come and remove the old gutters, then we had stucco/wood repairs and new fascia put on, then we painted, then the new gutters were installed. There was a lot of rot behind the old gutters, so I would def. remove the old before you paint because you might need to make some repairs. (Oh, and we had major roof work done too.)
Our new gutters are bronze, so we didn't want them painted (plus they are more decorative than typical gutters).
It's ok to not have gutters for a week or so, this way everything can be done thoroughly.
ponsonby