This inverted "U" pvc pipe configuration is located one or two feet from a water source/hose bib (see photo). It has a shutoff valve, but when I open it, I don't hear or feel any water flow. I'd like to cut and cap each end below ground level, but don't want to create any unnecessary problems. Any thoughts?
I believe you are describing a VACUUM BREAKER installation which looks something like this:
And NO, you cannot remove it and leave your system attached to the water supply. This device prevents backflow from the sprinklers back into the municipal water supply in case of a loss of pressure.
That is not a valve on top and it should not be adjusted without equipment and know-how. If you have changed it, you should get a sprinkler technician to come out and inspect it.