It was also the Irish Government that wanted to re-run the Treaty of Nice and Lisbon (they did it twice!). The Irish political mainstream was entirely behind Nice & Lisbon, and they were sorta suprised at the No vote. Hence they decided to re-run it.

The EU has no problem dropping a member state out if that government wants to not take part. The UK didn't want the recent Fiscal Compact, and was going to veto it. So the rest of the EU (bar UK & Czech Rep.) just went ahead with a different treaty.