Can the British Monarch (or the Crown) remove the Archbishop of Canterbury from office?

In theory, yes. The Monarch is the Supreme Governor of the Church of England, ("We acknowledge that the Queen’s most excellent Majesty, acting according to the laws of the realm, is the highest power under God in this kingdom, and has supreme authority over all persons in all causes, as well ecclesiastical as civil") and previous monarchs have removed archbishops. The methods used then would probably not be legal now, although there is a wide body of opinion that in this case they might be appropriate, failing which a study of Sharia law ought to turn up something interesting...

In practice, it would be a parliamentary decision like Acts of Parliament, and the monarch would sign on the line for the procedure to be carried out. The incident has fueled the fire of disestablishment - the movement in favour of ending the special status of the CoE as an official Church in England (not the UK - Scotland has its own established church, and Wales has no established church). The monarch could as part of the Royal Prerogative dismiss any bishop or archbishop - as the monarch has the right under Prerogative to appoint them. This would not be used without much consultation. The Church of England cannot by itself dismiss the archbishop.