You can now search for availability and book Aer Lingus flights using Avios on the Avios.com website ... although you probably won't want to actually book them there !
I've written previously about the high "carrier surcharges" that BA impose on european EI flights - and booking through Avios.com - they are even slightly higher !
Here's a few examples:
LHR - DUB 76 gbp
DUB - LHR 41 gbp
return 117 gbp
For context - if you chose a BA flight and had flown or earned miles in the last year (so were eligible for their lower fees reward flight) a single ex LHR would be 17.5 gbp and a single ex DUB would be 10.3 gbp (book through BA no Avios.com who charge everything at 17.5 gbp)
LGW - NOC 57 gbp
NOC - LGW 36 gbp (plus airport 10 eur "development" fee)
ex NOC return 94 gbp
ex LGW return 92 gbp
when I phoned up BA in June 2015 the NOC - LGW fee was 27 gbp
when I phoned up BA in August 2015 the LGW - NOC fee was 54 gbp
The longhaul prices are a bit more reasonable - but you still shouldn't book them through Avios.com
In the 13,000 avios category (for an off-peak economy single)
DUB - Hartford 100gbp
DUB - Boston 100 gbp
DUB - Chicago 109 gbp
DUB - JFK 100 gbp
In the 16,250 avios category (for an off-peak economy single)
DUB - Orlando 109 gbp
DUB - LAX 117 gbp
For a return flight the taxes and fees come to around 180 gbp - but a commenter over on HeadForPoints rang up BA and for the same flights the taxes and fees are 90 gbp
So ring up BA before booking EI flights on Avios.com