What Christian groups do not observe or celebrate the Eucharist? Are there any groups who identify as Christian but who do not celebrate/observe the Eucharist?
Two significant groups at least loosely associated with Protestantism that don't observe communion are The Salvation Army and the Quakers.

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.