Alexandria has an ordinance prohibiting use of metal detectors on public lands such as parks. If they were found at a grave site, it would be considered desecration of the grave if they were removed. If the artifacts were on private property, they belong to the property owner and it would be stealing if you took them.
Most artifacts are valuable in an historic sense only.
Our understanding of the past can make today and the future better. Even if you were to find an artifact worth some money, it is still against the law to take it if it was not found on your property. Artifacts tell us the story of the past. If you remove them, it is like having only one sentence from an entire book. Unless the site is actually threatened, archaeologists would probably choose not to excavate.