Can a US citizen marry an illegal alien whose visa has expired?

NO

need more details to answer this question. specify what kind of non immigrant visa are your talking about.

YES. If you do NOT have a valid visa to remain in the US, that alone is grounds for deportation. Given that the US does not currently keep track of whose visas have expired, however, there is some time leeway to find a solution to the problem of not having a valid residency visa. However, if you run afoul of law enforcement in any manner, and they bother to check your immigrant status, you can be deported solely for lack of a visa, even if you are never charged (let alone convicted) of any crime. Lack of a proper visa is itself a crime.

Recent changes by the Department of Justice allow for a non-citizen spouse of a US citizen to apply for a permanent resident visa EVEN IF THEIR CURRENT VISA HAS EXPIRED OR THEY ARE HERE ILLEGALLY. Once this application is made, the non-citizen does not have to leave the country while the application is being processed, and cannot be deported solely for lack of a valid visa. The non-citizen spouse will have to return to their home country to pick up the actual visa once it is granted, since they will need to re-enter the country on a valid visa to register that they are here legally.

Essentially, the new process is thus for the non-citizen spouse:

1. the non-citizen's visa status may or may not allow continued residence (or they may not have a visa at all)
2. The citizen and their non-citizen spouse apply for a permanent residency visa for the spouse based on marriage
3. The non-citizen applies for a waiver of the requirement to leave the country while the application for residency is pending. Prior to this year, this waiver was seldom granted. As of 2012, a change in the DOJ's viewpoint means that the waiver will generally be automatically granted.
4. Wait 6-18 months for the residency visa to be approved.
5. The non-citizen spouse must travel back to the country of their citizenship, and pick up their new US permanent residency visa at the US Embassy in their country (usually in the capital city)
6. The non-citizen spouse can now immediately return to the United States, re-entering with the now-valid US permanent residency visa.