The Aarhus Convention was drafted by the United Nations Economic Commission for Europe (ECE). It went into force in 2001 and was adopted by all EU Member States.  The EU had approved the Convention as an international treaty by Council Decision of 17 February 2005 (2005/370/EC). Article 216 (2) of the Treaty on the Functioning of the European Union (TFEU) provides : “Agreements concluded by the Union are binding upon the institutions of the Union and on its Member States.” Ireland ratified the Convention on 20 June 2012. So the Convention is a matter both of EU law and of national law in each Member State.
UNECE published an implementation guide, second edition 2013. 
The third pillar of the Convention is devoted to access to justice in environmental matters. Remedies against failure to act are explicitly provided in Article 9 (2)  and (3) . The EU transposed Article 9 by Directive 2003/35/EC, but not completely :  Article 9 (3) was excluded.
The Commission presented a proposal from 24 October 2003  for a Directive on access to justice in environmental matters transposing Article 9 (3) of the Aarhus Convention into EU law. This proposal did not get a majority in the Council.
The Court of Justice of the EU (CJEU) found in its judgment from 8 March 2011, case Lesoochranárske zoskupenie VLK (C-240/09), known as the “Slovac Brown Bear Case”, that Article 9 (3) Aarhus Convention has no direct effect. But it continued that it is for the national court to interpret, to the fullest extent possible , the procedural rules relating to the conditions to be met in order to bring administrative or judicial proceedings in accordance with the objectives of Article 9(3) of the Convention.
Case law on failure to act referring to the Aarhus Convention
Case law of the CJEU was not found.
Exhaustive data bases (in English) on national case law dealing with the Aarhus Convention do not exist. A data base is being established by UNECE.  In this collection three Swedish cases are worth being mentioned.
A company began to construct a windmill without obtaining an environmental permit. The County Board quashed the municipality’s “0-decision” not to prohibit the construction until the issuing of a permit. The Environmental Court of Appeal rejected the appeal of the municipality against this decision.
Neighbours complained about disturbances of a windmill. The “0-decision” of the municipality not to intervene was quashed by the Environmental Court of Appeal.
A neighbour complained about the operation of a landfill in breach of several conditions to the permit. He asked to revoke or update the permit. The supervisory authority found no reason to undertake any measures. The Environmental Court of Appeal revised its previous case law .It found that the neighbour has a right to appeal in such cases of operation without a permit.
The Aarhus Compliance Mechanism 
Article 15 Aarhus Convention provides :
“Review of compliance
The Meeting of the Parties shall establish, on a consensus basis, optional arrangements of a non-confrontational, non judicial and consultative nature for reviewing compliance with the provisions of this Convention. These arrangements shall allow for appropriate public involvement and may include the option of considering communications from members of the public on matters related to this Convention.”
At the first Meeting of the Parties (of the Convention) the decision I/7 on Review of Compliance  was adopted. .According to this decision a Compliance Committee (ACC) was established. Communications may be brought before the Committee by one or more members of the public concerning that Party’s compliance with the Convention. Meanwhile, there is extensive case law, which is published on the Internet. 
ACC case law
In this collection an interesting case on omission of public authorities was found (ACC findings and recommendations from 16 June 2006 ). In this case, which was pending in Kazakhstan, the communicants (neighbours) had filed various lawsuits against the operation of an industrial facility for the storage of cement and coal in the immediate proximity of the residential area where they live. Although an environmental assessment had stated multiple breaches of national environmental legislation the operation was not stopped by the public authorities. In the last judgement the national court had pointed out, that there is a right but no obligation to restrict or to suspend an activity. The court considered inter alia that imposing an administrative fine on the facility provided an alternative course of action for the public authorities in fulfilling their obligations. The Committee pointed out (par. 30 b) :
“The Committee is not in a position to interpret substantive
environmental and administrative legislation of the Party where it falls
outside the scope of the Convention, nor is it in a position to dispute
the court’s opinion that the public authority has a right to judge
which of the courses of actions available to it are best suited to
achieve effective enforcement. The Committee is, generally speaking,
reluctant to discuss the courts’ interpretations of substantive
provisions of environmental or other domestic legislation. However, a
general failure by public authorities to implement and / or enforce
environmental law would constitute an omission in the meaning of article
9, paragraph 3, of the Convention, even though the specific means
proposed by the plaintiff to rectify this failure might not be the only
ones or the most effective ones ; “
In my opinion the administrative procedure and the judicial procedure are to be regarded as a whole in such a case. The substantive illegality of the operation was not disputed. Article 9 (4) of he Convention requests effective remedies. So the extent of administrative discretion and the intensity of judicial review fall under the scope of the Convention.
The Committee found that the review procedure did not comply with the Convention in this case.
 “Each Party shall, within the framework of its national legislation, ensure that members of the public concerned :
(b) … have access to a review procedure before a court of law and/or another independent and impartial body established by law, to challenge the substantive and procedural legality of any decision, act or omission subject to the provisions of Article 6 …”
 “ In addition and without prejudice to the review procedures referred to in paragraphs 1 and 2 above, each Party shall ensure that, …, members of the public have access to administrative or judicial procedures to challenge acts and omissions by private persons and public authorities which contravene provisions of its national law relating to the environment.“
of 26 May 2003 providing for public participation in respect of the
drawing up of certain plans and
programmes relating to the environment and amending with regard to public participation and access to justice, Council Directives 85/337/EEC and 96/61/EC.
 COM (2003) 624 http://eur-lex.europa.eu/LexUriServ/site/en/com/2003/com2003_0624en01.pdf
 See Guidance Document on the Aarhus Convention Compliance Mechanism http://www.unece.org/fileadmin/DAM/env/pp/compliance/CC_GuidanceDocument.pdf ;
Koester, The Compliance Committee of the Aarhus Convention, in Environmental Policy and Law, 37/2–3 (2007) p. 83 http://cmsdata.iucn.org/downloads/cel10_koester.pdf
 http://www.unece.org/env/pp/ccmeetings.html and http://www.unece.org/fileadmin/DAM/env/pp/Media/Publications/ACCC_Jurisprudence_Ecoforum_2011.pdf (summary of the case law 2004 -2011)