Are there any groups who identify as Christian but who do not celebrate/observe the Eucharist?
The rationale for the Salvation Army's position is more fully explained here: Why does the Salvation Army not administer the sacraments of baptism and the Lord's Supper? But the short version is that they don't observe the Lord's Supper because (1) it has been a source of division in the church, and (2) it is not required for salvation.
The rationale for Quakers focuses more on all of life being "sacramental"; see, for example, this summary from the BBC:
[Quakers] don't regard some activities as more sacred than others, nor do they believe that any particular ritual is needed to get in touch with God, so they do not believe in the sacraments practised in mainstream Christian churches. (source)
As for Messianic Jews, their practice varies on the actual denomination that they belong to. Some tie communion closely to the annual celebration of passover, while others observe it regularly like other Protestants.
There is another group who call themselves Christian, but who eschew being identified either with any kind of Catholicism or Protestantism, and they avoid the word 'celebrate', preferring to speak of 'commemorate'. There is only a tiny percentage of their members who partake of the elements, the rest believing that this memorial event is only for them to look at.
They restrict the event to once a year, and they call it the memorial of Christ's death, a solemn event and not a celebration. The word 'Eucharist' is never used by them either and they certainly don't believe in transubstantiation. They are the Jehovah's Witnesses. Up until 1935, virtually all of them took the bread and wine at their annual memorial event. That was because they all believed themselves to be in the new covenant Christ inaugurated at the Last Supper, and that they were called to the heavenly kingdom. However, a new teaching had arisen, claiming that only 144,000 such persons would ever be in that new covenant and go to heaven. This had a direct bearing on restricting the numbers partaking of the bread and wine because those not in that tiny group were said to have an earthly hope, and were not in the new covenant. They were not entitled to partake of the emblems.
At their Washington D.C. convention on 31 May, 1935, it was announced that “anyone coming into the organization from that day on would compose an earthly class of Christians.” [See their history book, 'Jehovah's Witnesses Proclaimers of God's Kingdom', pages 166-167, published 1993, also 'The Watchtower' magazine 1st & 15th of August, 1935.]
Jehovah's Witnesses claim the 'other sheep' class began to be gathered particularly from 1935 onwards. They also believed that the 'little flock' class had almost reached its numeric limit of 144,000 members shortly after then. All Jehovah's Witnesses claiming to be part of the little flock said they were convinced they would go to heaven. In 1935, memorial attendance worldwide was 63,146, and 52,465 partook, testifying to their heavenly calling. From 1938 new members who were said to be 'other sheep' were invited to attend the memorial but not to partake. 1938 saw 73,420 attend but only 39,225 partook. That showed a drop of 13,240 claimants of the little flock in only three years. Significant drops continued right through to 1995 - it was down to 8,645 partakers then. By 2015 it was up to 15,177.
Yet given there are more than eight and a half million members, it still remains true that only a tiny percentage of them feel entitled to partake of the bread and wine at their annual memorial event. There are thousands of congregations where not one single member partakes. There is no observance of the event by the vast majority who just pass the plate, then the cup, to the person next to them, in silence. Perhaps they would call that an 'observance' but that does not seem to be the sense of the word as you use it, in your question.