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.
While the other answers talk about using your current tires for use on a trainer, I'll take a bit of a different route. Tire manufacturers make special tires for specifically trainer use, such as the Continental Hometrainer (plenty of other manufacturers make similar ones, but the pricing locally for this one isn't bad - they typically run 25-35 dollars each if you shop around). These prevent you from wearing out your road tires prematurely (especially since only your rear tire wears on a trainer), are quieter and are more durable on trainers than most road tires (but are NOT suitable for off trainer use). You can run them up to the pressures suggested by the manufacturer on the sidewall or how much your rim can bear or possibly a bit more (though these are not necessarily optimal - play with them) - since there are no bumps or terrain on a trainer, you don't need to worry about overly bouncy tires or pinch flats or other hazards. See the link from Sheldon Brown in the other answers. I'd recommend buying a tire designed for trainer use, putting it on the rear, running it for the winter months, then swap it out for the usual tire when spring comes along. This should reduce your tire expenses over a few seasons.
I would recommend the normal tire pressure, because lower pressure cause unneeded vibrations, noise, tire slips over the trainer, tire excessive wear, and also different resistance if you using speedometer as a powermeter.
Interestingly my road tyre has a small imperfection (possibly from being sat on the trainer in the cold?) that manifests itself as a vibration at 17mph-> causing a significant 'wall'. I could see the roller (Kurt Kinetic) vibrating. I played around, first checking the trueness of the rim and then reseating the tyre (tread is near perfect as it's that new), but then I found that dropping the PSI from 100 down to 85 and slightly tightening the roller made a significant difference - a bit less PSI and it got better still. I guess that the tyre is now deforming more readily and causing the roller to move in/out less. I do have a spare wheel (needs a bit of a rebuild) and I'm tempted to go for a trainer tyre myself.