So, what is the point of a church altar?
In Roman Catholic liturgy specifically, the altar is where the Eucharistic Sacrifice is made; i.e., Christ's one sacrifice on Calvary is made present again on the altar at each mass. The altar, in this theology, is still for performing sacrifices.
"In the New Law the altar is the table on which the Eucharistic Sacrifice is offered. Mass may sometimes be celebrated outside a sacred place, but never without an altar, or at least an altar-stone."
As the largest unified body of Christians on the planet, Roman Catholics have a large influence on the terminology we use. Note that as a Protestant I don't agree with this theology, but the term "altar" likely persists in our churches merely as a matter of "stale" terminology used by those who do not know better. Protestant theology (at least all that I'm aware of) does not allow for an "altar." The reason to have an altar is antithetical to the Protestant doctrines regarding regeneration and justification and having something that fulfills the purpose of altar (meaning that it isn't just a matter of stale terminology) is nonsense in Protestant churches.
In Eastern Orthodoxy, the Altar is the type of the Altar in Heaven as described in Revelation. In the Anaphora, the Church joins in the 'One Liturgy' going on in heaven, and the Christ who offers himself once and for all becomes present in the gifts on the altar as a bloodless, spiritual sacrifice.
As a secondary note, it is traditional for the relics of Martyrs to be embedded in the bottom of the altar, following Revelation.
In the church in which I grew up (a United Methodist church), we have a wooden altar in the middle of the front of the sanctuary. It serves a purpose in two main contexts in the service:
The offering plates are laid on it after the offering is received, and the pastor prays over them.
During communion, the bread and wine are kept on the altar, and a short rite involving breaking the bread is conducted before communion is taken by the congregation.
In light of these two things, Here are my thoughts on the altar, with the caveat that none of this is confirmed as doctrine:
The altar is an ancient religious symbol of sacrifice. I agree that the Jewish laws of sacrifice are not demanded of Christians, and in light of that alone it seems strange to have an altar any more. However, as Narnain mentions, the altar is very different now than it was then. So if it's not a Jewish sacrificial altar, what is a Christian altar?
My answer is that it's still a symbol of sacrifice, but a whole different kind - it's a symbol of Christ's sacrifice. It's an altar that has had its order filled, and we keep it as a reminder of that sacrifice. Likewise, whether they are mandated in the same way or not, Christians still make sacrifices and offerings to God, and I find it very fitting to make such a sacrifice on an altar that is symbolic of Christ's sacrifice for me.
I don't believe there is a scriptural demand for churches to contain altars, and neither do I believe that there is scripture which explicitly disallows altars. I feel like they are much more symbolic than anything, but I think there's plenty of grounds for justifying one.
An altar is not an altar if it hasn't been made as a place for sacrifices, but we can conclude it is still necessary. Notice that Hebrews 13:10 says, 'We have an altar...', and if food were not offered there, it needn't have been said that Levitical priests were barred from eating at it, so to not have one and not offer the required food is to stand apart from the author and from the first Christian Hebrews. Thus we can conclude sacrifices of some sort, including of food, are still necessary.
What really is the antecedent of 'It' in 'It is finished'? One might say the Sacrifice is the Sacrifice to end all sacrifices, but with what basis? Or one might say it is for all time the one Sacrifice for sins, concluding it is the Sacrifice to end all sin-offerings, and no more animals are to be sacrificed.
But other sacrifices have not been abrogated: the grain-offering (Notice, instead of reproving the Jews for the understanding that He required them to eat His flesh and drink His blood, Christ repeated Himself about it even more plainly.) and the drink-offering, so bread and wine are offered, and incense (Malachi 1:11 and Revelation 8:3,4) as well. (Of course it is commonly understood the sacrifice of praise, offering bodies as a living sacrifice, and offering of money continue too.)