Nice thread! Good questions, neat answers!
I'm curious as to the traditions With Orthodox altars.
-Do Orthodox altars typically have relics placed in them?
It is traditional, yes. When a church is consecrated, particularly the Holy Table itself, relics are placed inside. Many parishes in the US, though, don't have permanent altars (such as my parish, which is currently in a rented space) and therefore our altars do not have relics in them. An antimension (Gk., literally "instead of the table"), which is a cloth originally used to serve Liturgies on the go (i.e., for military chaplains) are now standard for serving liturgies, and always have a relic sewn into them. They are also signed by the diocesean hierarch. Though, I've heard some churches with a consecrated altar use another type of cloth that doesn't necessarily have relics. Forget what those are called, but all antimensia have relics.
Placing relics in Holy Tables and antimensia goes back to the practice of the early Church that served the Liturgy on the tombs of the martyrs, a reminder that Christ is "trampling down death by death and upon those in the tombs bestowing life."
-Are they typical of a particular shape? (I think I've only ever seen square, as opposed to the western rectangle)
Square is the most traditional, I believe. As one poster said, it's in remembrance of the Ark of the Covenant, on which the sacrifice of blood was sprinkled once a year for the sins of the people (Yom Kippur), now it is the place of the once-for-all sacrifice of Christ in the Eucharist. Ideally, the Holy Table is the exact dimensions of the Ark, which are given in the Pentateuch (in cubits, IIRC). Of course, this was also mentioned earlier.
I had never heard that western altars are so shaped in order to resemble a tomb. Just...never thought about it. Quite interesting!
-Do the Orthodox make use of a tabernacle? Or is this impractical due to the presentation of the Eucharist?
Yes, there should be a tabernacle on every Orthodox Holy Table that contains the Reserve Sacrament. It is replaced every Holy Thursday at the Liturgy of the Mystical Supper (which is why the Communion of the Clergy seems to be so long during that Liturgy, the priest is preparing the reserve Eucharist!) The remaining Reserve Sacrament is added to the chalice for the Communion of the Faithful and consumed. Also, Reserve Sacrament can be set aside at any Liturgy, doesn't have to be on Holy Thursday. This is particularly the case if a parish runs out before the next Holy Thursday comes around! Though, that is the traditional time it is done.
The Tabernacle sits on the Holy Table, and is almost always under glass. Many times it looks like a small church, as the picture given by another poster above indicates.