I'm interested in the opinion of the Catholic Church, but any others will be also appreciated.
The Catholic Church directly contradicts many of the teachings of Swedenborg, but as best I can tell has not directly addressed Swedenborg himself. I am unaware of any documents from the Magisterium directly addressing Swedenborg. Thus, in that sense, the Catholic Church does not have an official position towards Swedenborg. However, many of Swedenborg's teachings directly contradict the doctrines of the Catholic Church and so would be condemned as heresy.
So, to state it a different way, we cannot talk about the Catholic position on Swedenborg as a whole (because one has not been taken), but we can talk about the Catholic position on his individual teachings. It is completely possible that some of his teachings are in direct alignment with the Catholic Church while others are heresy.
Let's take a look specifically at the teachings listed in the Heaven and Hell article. I've attempted to be brief and so only touched on the major points. Please feel free to ask questions on anything I didn't cover and I will do my best to answer.
Swedenborg teaches that angels used to be men and women and still live quite a bit like humans on earth. This, again, is a direct contradiction to the Catholic teaching. The Catholic teaching is that Angels are pure spirits (no corporeal bodies whatsoever) and are a separate nature from human nature. Angels were created separately from humans and humans never become angels.
Both Swedenborg and the Catholic Church agree that marriage is between one man and one woman and "what God has joined, let no man put asunder." However, they disagree on the nature the marriage bond.
Heaven and Hell
The first difference between Swedenborg and the Catholic Church is the nature of judgment after death. Swedenborg teaches the concept of a "World of Spirits" in which a person's "true nature" is revealed over time after they die. Everyone enters this World of Spirits at death and over time whether or not Heaven or Hell is a better match for their nature is revealed to them. They then naturally choose the better match.
Catholic teaching is that when we die we are immediately judged and enter Heaven, Hell, or Purgatory. There is no reveal over time of our "true nature" but a direct judgment. Those who have rejected God are sent to Hell and those who have not attain Heaven. (Possibly going through Purgatory on the way to Heaven. Purgatory is a place for those who have gained Heaven, but first need to purge a remaining attachment to some earthly thing).
There is an interesting distinction between these two viewpoints. In Swedenborg's teachings, those in hell don't want to be in heaven. In the Catholic teaching, the desires of those in hell are completely irrelevant, they simply cannot enter heaven. They have rejected God and so cannot attain the beatific vision (see below).
The other major difference appears to be in the definition of heaven and hell itself. The Catholic Church defines Heaven and Hell primarily in terms of the gaining or the deprivation of the beatific vision. That is, the beholding of God's glory and seeing him face to face. Heaven is the attainment of the beatific vision, Hell is the deprivation of this glory. While the Catholic Church does teach that we will be active and doing "stuff" in heaven, the focus and definition is on the beatific vision as that is the greatest glory and dwarfs all other occupations of heaven.
Swedenborg thought of heaven, not as a beatific vision of God, but as a realm in which people live in community and engage in acts of love and service toward one another. His vision of heaven was closer to that of a good society here on earth, though in a spiritual rather than a material way. He described various occupations in heaven, most of which today would be considered "service industries," such as education, child-raising, preaching, and so on. Though he did describe periodic worship services in heaven, he rejected the idea that angels spend eternity in unceasing worship, praise, and contemplation of God.
[Update]: Rewrote answer on advice from Lee Woofenden
The main issue the Catholic Church will probably have with Emanuel Swedenborg is that the doctrines revealed to him does not describe the Trinity as three distinct persons, but rather as different aspects of the Divine in the one person Jesus Christ. According to the revelations, the Trinity of three distinct persons was not known in the early church, and the apostles would have been surprised to see how Christianity developed in the way that it did. According to Swedenborg, the Trinity of three distinct persons was an invention of men to counter Arianism (that Jesus was a created being), but in so fighting against the Arian heresy they put something just as bad in its place.
As for the above comments on marriage, that is in fact a minor point, as Jesus was probably talking against the belief of an actual physical resurrection in a regular human body. Moreeover, at death many couples do separate, in the afterlife many find their true "soul mate."
Other than that, the Catholic Church may attempt to remain "neutral" on Swedenborg, as the revelations given to him speak out strongly against the "faith alone" doctrine of the Protestants. His theological writings show in detail that the Protestant reformers not only misinterpreted the writings of the apostle Paul, but this doctrine was made central by the Protestants to help ensure that their churches would remain separate from the Catholic Church. Swedenborg does speak favorably, however, on their making scripture the source of spiritual truth.