m134e5:
Absolutely NEVER
This question isn’t very clear- you should be more mindful of punctuation. I’ll answer it as best I can. “If so, why?”- because the Koran is not inspired by God, and Islam is not compatible with Christianity. “Is it against the Catholic faith to take part in a non-Catholic worship?”- depends on what you mean by “take part”- if you just go to observe- it is fine, unless it endangers your faith. “Why is it against the Catholic faith to take part in a non-Catholic worship”- well, if you participate in their rituals, you are professing a Faith that is not Catholic.
The problem here is that the Vatican has, apparently, no difficulty in principle with such activities.
And yet, it used to be dead set against them - to put it mildly. Which seems to show that the Vatican is very good at justifying completely opposite courses of action. Which is a good way to undermine confidence that the Vatican knows what it is doing: for it can hardly be appalled at the notion that Protestants should receive the Eucharist (as it used to be) - then behave as though it were no big deal for Protestant ordinations to take place in Catholic cathedrals, or for Catholic churches to be the place for Sikhs, Muslims, Jains, Buddhists, etc., to worship.
It is not enough, theologically, to say that those in authority have authority to alter the laws which govern how the Church behaves: for though that is true, the authority of the Church to legislate can’t be unlimited - it is limited by the mission and nature of the Church: so the appeal to law is not an appeal of unlimited validity: which is why no law could justify (let us say) human sacrifice in a church: for human sacrifice is not something the Church can ever allow as an act of worship - so it can’t be legislated for.
That is an extreme example

- it shows that some things cannot be within the competence of the Church to do, not doctrinally, not legally either.
And ISTM that the Church cannot justify, legally or otherwise, acts of inter-religious worship.
Apart from anything else, what is the theological basis for such things ? They need to be shown to be an expression of the Church’s worship of the Blessed Trinity - but how can those who do not have faith in Christ witness to that Mystery if they are not part of the community of the redeemed, the Church of Christ ?
Christian worship is a public act - it is an act of believing Christians: so how can those who are not believing Christians, share in it ? God is not confined to the Church - but neither is the Church the same as the human race: to be Christian, is not compatible with animism, or nature worship, or polytheism. So there are acts of worship which are not acts of Christian worship. There are things that Christian worship is - and things that it is not. There are things that it cannot be, cannot include, things which, if included, cannot be made Christian.
Muslims (say) may be “worship[ping] the Father in spirit and in truth” - but if so, that is known to God, not to us in the Church. So Muslims cannot take part in the Eucharist. because Christian faith is a social thing, made visible by acts, words, sacraments, and so on. Since they do not do anything which is specifically Christian, their worship can’t be assumed to be Christian - it has to be assumed not to be.
(BTW - “the Vatican” is my shorthand for “the characters who organise and teach and regulate the official expressions of the life of the Church on earth” - for we are all members of the Church, just are they are: so calling them “the Church” is insufficient.)