A reentry ban issued by the UK does not affect movement in or through the Republic of Ireland in a strict legal sense. Ireland is a different country and hence outside the UK's sovereignty. So whatever happens falls to the Garda Síochána to decide.
Having said that there's an open data sharing between the UK and the ROI (and by proxy the USA). And the Garda may decide that your fiance's UK ban makes passing through the ROI an unacceptable absconding risk.
Your fiance has noticed that flights passing through the Republic are generally cheaper.
Would they turn him away when he arrives there?
This is unlikely. A more likely scenario is where your fiance is flagged up in the Advanced Passenger Information scheme and he will be unable to board the aircraft. This would not be from the ban itself, but as explained there's a high absconding risk which the Garda may decide is unacceptable.
Are there exceptions if its a flight transfer/layover?
There are no exceptions of any help to your fiance. However nothing prevents your fiance from applying to the Irish Naturalisation and Immigration Service for a transit visa. Doing so avoids the chances of wasted airfares and the distress of a denied boarding.
Please help! Thanks
Given that a ban is in effect within the EEA, your fiance should absolutely apply for a Schengen Short Stay Visa (Netherlands variant) before leaving. Alternatively you may wish to effect a rendez-vous in the US.
Note: User Calchas (to whom thanks) strengthens this answer by providing a relevant entry from the Travel Information Manual Automatic (TIMATIC)...
"Additional Information: - Visitors holding passports containing a
British inadmissible stamp could be refused entry."
Note that the language uses modal verbs which means the Garda have the option, but not the obligation, to refuse entry (or boarding permission). So it is my own conclusion that they will refuse, even though the ban was issued in a different country.
The source for this quote is pay walled and cannot be accessed. A comparable substitute is here.
An observation from Crazydre (to whom a tip of the hat) that Schengen visas are not available to Americans...
You cannot get a Schengen short-stay visa as a
...prompted an email enquiry to Elspeth Guild (who teaches the course on Schengen visas and a personal acquaintance for over a decade), and she replied...
...in practice US nationals do sometimes apply for Schengen visas and
the applications are processed in the same was as if they were visa
nationals. Does this help?
So the information that has been previously vectored on the net is likely misconceived or confused or outright wrong. It's a problem with visas and the internet in general. Visas are an emotive topic and it's actually a challenge to get to the bottom of things because people vector incomplete or inaccurate information. Don't use the net when something's important.