Are there any denominations that teach a Hell that is not a place? To be more specific (and also more broad), are there any major groups within Christianity that consider Hell not to be a specific place, but rather a specific state of being?

I expect the list to be short, so this shouldn't be a bad list question.
This seemingly simple question is going to require a relatively complex answer.

A good place to start is this Wikipedia article. As it shows, there is Christian disagreement on whether Hell is primarily a state, or primarily a place, and that disagreement is not necessarily along denominational lines. Even those that describe Hell as a place may consider that language symbolic: Pope John Paul II stated on 28 July 1999 that, in speaking of hell as a place, the Bible uses "a symbolic language", which "must be correctly interpreted … Rather than a place, hell indicates the state of those who freely and definitively separate themselves from God, the source of all life and joy.

Most Evangelical Protestants, asked to define Hell, would usually say "Hell is separation from God", rather than define it as a place where torture and pain occur: eternal separation from God is sufficiently painful that no other punishment is necessary, and the language of fire and brimstone is simply a picture of that pain (an atheist would of course contend that separation from God is no punishment at all). Billy Graham defines Hell in that way in his seminal evangelical book Peace with God.

Although the definition of Hell is separation, that does not necessarily imply that Hell is not also a place. For separation to be real there must be no interaction, and it is therefore implict that those separated from God (and others who have chosen to be with him) are in 'a different place' in some sense. However for mainstream Christians the 'being in a different place' is secondary.

There are significant numbers of Christians who would disagree with this view - even some that literally believe Hell is at the centre of the earth - but since they frequently do not divide along denominational lines, any attempt to produce the list the question asks for is doomed to failure.

A good introductory read (which should itself be taken as allegory rather than a theological treatise) is C.S. Lewis The Great Divorce. A simple Google search will also turn up all sorts of views on Hell and its nature.

You should also read the answers to this question.