Anyone know what taxes would be after the abatement is gone?

Short answer is no, but you should be able to ask for an estimate from the broker. Otherwise, you best bet is to mine the city website and do your own calculations. Given the prices of those apartments and in Tribeca generally, bet high.
http://www.nyc.gov/html/dof/html/property/property_condo_coop_comp_rental.shtml

I have l;ooked at a great deal of properties in Tribeca as well as in Chelsea and the West Village. I am not sure what the taxes will be when the abatement expires,but isn't that in 10 years? I can tell you unequivocally that the Fairchild's current taxes are significantly lower than other similar properties with abatements.

If your feeling sleuthy and it's important enough to you, propertyshark should list the total preabated tax for the building somewhere on the building page.
Then find the unit your interested in, look it up on acris and fish through the condo declaration to find the percentage of ownership this unit has.
For example, if you see the building tax total is $300,000 and the unit's ownership percentage is 8.4%, the unabated tax for the unit would be $25,200 for the year or $2100 per month.

And for a short cut neighborhood of cost for unabated taxes, the few times Ive gone through the trouble of researching, I have found the abated amount represents @ 10% of the unabated tax.
SO for example, 4C in contract showing $290 in tax, may likely be around $2900 per month.
But I offer no guarantees or accuracy for this calculation.

Ok, decided to give it a go, it was much quicker to do then I remember;
Ok here is the propertyshark total tax

(On propertyshark) Annual Tax Bill $491,989 / Square Foot 45,000 SqFt / Tax per SqFt $10.93

So the building assessed tax bill was $491,989

(On acris, condo declaration page 28)4C has 5.8142% common interest)

5.8142% of $491,989 equals $28,605.22 per year or $2,383.77 per month

Of course that's today, not ten years from now.
But I'd expect taxes to rise near the same proportion against the common charges for the unit.

Ultimately,like with most 10 year tax abated properties, probably to sell by year 7.