yes, they should be closed...but that won't affect the pump at all. if the valves are open, the pump should still pump water...it just won't come out of the faucets. it'll be pumped out the drain valves and onto the ground.
When I took possesion of my trailer, the original pump ran, but wouldn't pump anything out of the tank. I figured it would be easier to just replace it, so I did. in the process, I found that the plastic line from the water tank to the pump had a split in it, and had to be replaced, as well. that original pump might actually work..(probably needs new seals/diaphram...hadn't been run in years), but it was just sucking air. and I didn't spot this problem at first, because its so dark under that cabinet, and the line was black, plus its a "suction" line, so nothing would leak out of it anyhow....
'73 Safari 23'
'00 Dodge Ram 1500 4x4 QC