Does tire pressure matter when the bicycle is in the trainer? I keep my bicycle on a trainer in a 'cool' room during the winter months.   The tires don't seem to maintain pressure after days of non use.   Does this matter, and should I be checking/filling the tires often?   What pressure would be recommended?
Like car tires, bike tires can lose some pressure at colder temps (see: Why do tires "lose" air during the winter?) although I'd be surprised if your 'cool' room is cold enough for this to be a significant factor.

More likely, you are just seeing the normal slow loss of air pressure that bike tires experience all the time. 

Since your trainer is a relatively "controlled" environment compared to a road, tire pressure is of less importance than it is on the road.  If your tire is under-inflated you really don't need to worry about a pinch flat or the tire coming off the rim as you corner.  Conversely, if it's a little over-inflated, you won't suffer bone jarring bumps or benefit from lower rolling resistance.

At worst, I think you might experience some accelerated tire wear.  Of course, my trainer seems to eat tires no matter what pressure I run my tires at...

Sheldon Brown has a nice write-up about Tire Width and Pressure which you might find interesting even though it doesn't specifically discuss tire pressure when using a trainer.

Edit:  I overlooked the What pressure would be recommended? part of the original question...  

To me, the answer to this depends on what tires you are using.  My current tires have a maximum pressure of 120psi and I run them at around 100-110psi.  In the summer I check my tires and re-inflate before every ride due to the various reasons I've listed above.  In the winter, I only check the tires occasionally as there is minimal drawback to riding a trainer with under-inflated tires.