These two may work as friends, but their natures are at odds in a way that leads quickly to trouble. The eager Aries might stun the reserved Virgo at first with their upfront, at times unsubtle ways. Some born under the sign of the Virgin have reserve with dating, and might like an old-fashioned courtship, with incremental moves.
Virgo is particular, and in some ways hard to get to know well. They will judge a lover by their actions, not their image. Aries' fervor makes them like a Bull in a China shop, since Virgo is one of the most sensitive (physically) signs. Many Virgos are high strung and private, and Aries can feel like too much, too soon.
Virgo may seem like a mysterious, self-contained loner that is hard to attain. But Aries loves to dash off for a spontaneous adventure, setting alarms off for Virgo about the details -- where will we stay, how much will this cost, etc. The Aries flames are smothered in all the Virgoan fretting, analyzing and cautious planning.
Virgo sees Aries as reckless and missing crucial steps, and Aries will feel ensnared in a web of criticism. This goes around in a vicious cycle until Aries storms out the door, not looking back. Unless there are other compatible planets, this is a mismatch that makes each feel insecure.
Keep in mind that Virgo needs a certain amount of stability, and for things to unfold over time. The sensual nature of Virgo is revealed in stages, and it takes patience to lure it out. Virgo warms up to someone who is there, day in and day out.
Aries likes living at the edge of experience, taking risks, throwing caution to the wind. The restless Aries might even see disinterest in Virgo, or make a snap judgment that they're all work, no play. Virgo simply needs particular conditions to feel at ease, while Aries thrives in new situations where there's the tension of new challenges. These two ships should pass in the night and look for more hospitable harbors.
See also I'm a Virgo in Love with an Aries -- "Help, What Should I Do?"