Is a Crucifix Necessary for Mass?

Well, the short answer is YES!  According to the GIRM:
II. ARRANGEMENT OF THE SANCTUARY FOR THE SACRED SYNAXIS (EUCHARISTIC ASSEMBLY)

308. There is also to be a cross, with the figure of Christ crucified upon it, either on the altar or near it, where it is clearly visible to the assembled congregation. It is appropriate that such a cross, which calls to mind for the faithful the saving Passion of the Lord, remain near the altar even outside of liturgical celebrations.

On the USCCB site we find this:  
the revised Instruction speaks always of “a cross with the figure of Christ crucified upon it.” (308, 122) This cross, “positioned either on the altar or near it,” should be clearly visible not only during the liturgy, but at all times recalling “for the faithful the saving passion of the Lord, [and] remain[ing] near the altar even outside of liturgical celebrations.” (308) A liturgical procession is a sign that the people of God form the pilgrim church on earth, it is fitting that such processions should be preceded by the Cross.

ROME, DEC. 20, 2011 (Zenit.org).- Answered by Legionary of Christ Father Edward McNamara, professor of liturgy at the Regina Apostolorum university.

 Q:   My parish has a beautiful crucifix mounted on the wall behind the altar that has been a great aid in my prayer life. Unfortunately, I must pray without this aid during the seasons of Christmas and of Easter, as during these seasons the crucifix is completely covered. During Christmas, a star is placed above the crucifix with a tail that hangs down to completely cover it. Likewise, during Easter, a banner of the Risen Christ is hung over the crucifix so that it is hidden from view. I realize that "a cross, with the figure of Christ crucified upon it, [be] either on the altar or near it" during Mass (General Instruction of the Roman Missal, No. 308), and I considered the processional cross, which is placed beside the sanctuary during Mass, to fulfill this requirement when the crucifix behind the altar is covered (cf. GIRM, 122). However, upon further reflection, I now question if the processional cross fulfills this requirement as it is beside the sanctuary during Mass and not "next to the altar" (GIRM, 122); and, it is not "clearly visible to the [entire] assembled congregation" (GIRM, 308). As well, it does not "remain near the altar even outside of liturgical celebrations" (GIRM, 308). Is it appropriate that the crucifix mounted on the wall behind the altar be covered during any liturgical season? -- R.G., Leduc, Alberta
 A:   While I don't think it is a good idea to cover the cross during these liturgical seasons, it does not appear to be illicit.
It is illicit, however, not to have any crucifix presiding over the altar during the celebration. The processional cross could fulfill this function, but only if it is placed on a stand beside the altar during the celebration.
Indeed, the indications in the norms referenced by our reader are that the processional cross is only carried out of sight in those cases where a crucifix is already present on or near the altar. If there is no cross, then it should be placed near the altar and serve as the altar cross.
Another possible solution, if the wall cross is covered or absent, is to place a crucifix upon the altar proper. In this case the processional cross should be carried away to one side so that only one cross presides over the altar.
While there may be no absolute prohibition to substituting the main crucifix for a smaller one during these liturgical seasons, I am of the opinion that it is not a felicitous idea.
As the U.S. bishops' conference recommends in its document "Built of Living Stones":
"§ 123 § The tradition of decorating or not decorating the church for liturgical seasons and feasts heightens the awareness of the festive, solemn, or penitential nature of these seasons. Human minds and hearts are stimulated by the sounds, sights, and fragrances of liturgical seasons, which combine to create powerful, lasting impressions of the rich and abundant graces unique to each of the seasons.
"§ 124 § Plans for seasonal decorations should include other areas besides the sanctuary. Decorations are intended to draw people to the true nature of the mystery being celebrated rather than being ends in themselves. Natural flowers, plants, wreaths and fabric hangings, and other seasonal objects can be arranged to enhance the primary liturgical points of focus. The altar should remain clear and free-standing, not walled in by massive floral displays or the Christmas crib, and pathways in the narthex, nave, and sanctuary should remain clear."
In the case described, the crucifix as an important, albeit not primary, liturgical point of focus is obscured rather than enhanced.
While a star is a frequent symbol of Christmas, and even of Christ, placing it right behind the altar places too much emphasis upon a secondary symbol.
While the figure of the risen Christ might appear more justified, nothing would be lost and much gained by placing the image in some other part of the sanctuary.
I hope that this practice is not an attempt to deliberately remove the crucifix from sight during these seasons. This would be a grave error. The Church insists that a crucifix must always be present for Mass during all seasons of the year in order to remind us of the presence of Our Lord's infinite sacrifice.
It is through the infinite sacrifice that Christ's entire saving mystery, from the annunciation to the ascension, is made present in each and every celebration. Even though we designate certain times and seasons to underline specific mysteries, the cross remains at the heart of the mystery of God's total self-giving for our salvation.

Addendum 1/5/2014:
When I was first asked about this the actual question was about "validity" of a Mass without the crucifix.  My findings made it clear that while there is a requirement in "law" (GIRM 308) there was no statement regarding validity - so I did not make a statement regarding validity initially.  My assumption was then, based upon this evidence, that Mass celebrated without a crucifix present would be illicit, but not "invalid."  The matter of validity would be centered upon the Sacrament of the Eucharist.  Regarding the Eucharist, if matter, form and intent are correct - it is a "valid" consecration (such as the Masses celebrated by SSPX, which don't have explicit permission of the local bishop, are "valid" but "illicit.")  A Eucharist which is valid, but illicit still does suffice for satisfying ones "Sunday Obligation."  So even if there is no crucifix present the Eucharist itself may be valid.

This same question was asked of Catholic Answers and while they do not provide reference material for their answer beyond the GIRM (they cite 270, which is an older version of the GIRM, in the current version it's 308 - as quoted above), their conclusion is the same as mine:

Must a crucifix be present at the altar for the Mass to be valid?

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Full Question

I would like to know if it is necessary for a crucifix to be present at the altar in order for the Mass to be valid.

Answer

Masses are not valid or invalid, they are licit (in conformity with the law) or illicit (not in conformity with the law). It is the consecration of the Eucharist within the Mass that can be valid or invalid.
Whether a crucifix is present at the altar has no bearing on whether the Eucharist is valid or invalid (the absence of a crucifix will not cause the Eucharist to be invalid), but it does have a bearing on whether the Mass is licit or illicit.
The law requires that "There is to be a cross, clearly visible to the congregation, either on the altar or near it" (General Instruction of the Roman Missal [GIRM] 270). The revised General Instruction—which has not yet gone into effect—clarifies that the cross in question should have a corpus (representation of Christ’s body), meaning that it should be a crucifix rather than a bare cross.
If there were no cross by on or near the altar (or, once the new GIRM goes into effect, no crucifix) then the Mass would be illicit, or not celebrated in accord with the requirements of the law.
However, a Mass celebrated in this manner would still have a valid consecration of the Eucharist. Furthermore, it would still fulfill one’s Sunday obligation.

I hope this helps...

Scott<<<