The same question was asked, on this Vaishnavaite website. I am just reproducing the same answer that was provided there.
Rudra Hridaya Upanishad is inauthentic because it contradicts many statements in the Vedas that declare Mahavishnu as supreme and aprAkRta (eg., Narayana Upanishad, Mahopanishad, Subala Upanishad, Narayana Suktam, Purusha Suktam, etc. etc.). This also contradicts many statements in the two itihAsas -- Srimad Valmiki Ramayana and Mahabharata -- where there are innumerable statements of Vishnu's supremacy over Brahma and Siva.
We can reinterpret these same “Upanishads” in a manner that actually refutes their opinions!
Note: These Upanishads are indeed bogus and must be discarded as pramAna. We are only interpreting them as a establishment of our stance and to close off these arguments once and for all. That way, even spurious texts will not be used to give distorted meanings of shAstra.
“aa umaa saa svaya.n vishhNuh |” (text “Rudra Hridayopanishad” 1:05)
Shaiva/Shakta interpretation: “Uma Herself is in the form of Vishnu”
Vaidika Interpretation: Fame or Yashas (Uma) is verily VishNu. “Uma” means “yashas” and is the ability to grant brahmAnandam or the ability to mingle with those lesser than himself.
sarvadevaatmako rudraH sarve devaaH shivaatmakaaH .
rudrasya dakshiNe paarshve ravirbrahmaa trayo.agnayaH .. 4..(text “Rudra-ridayOpanishad”)
Vaidika Interpretation: Narasimha who is the destroyer of the disease of samsara (Rudra) is the inner self of all the devas, and the devas are always immersed in bhakti (of nArAyaNa) of an auspicious nature (shivAtmaka). On his left (skilful) side are the Sun like eye, the quality of greatness (Brahmatva) and Sacrifice symbolised by the three agnis.
Purusha Sukta says “ahO rAtrE pArshvE” – day and night are at his sides. In other words, bhagavAn nArAyaNa reconciles contradictions. He has both grace and anger, a snake and an eagle as his servants, etc.
So, here, the mantra describes his two sides. Starting with dakShina pArshvE, it talks about his three characteristics by which he is terrifying to the asurAs. “DakshiNa” means left, but also means “skilful”, so it could be a play on how his actions are both potent and skilful. The 3 characteristics of “vinAshAya ca duskrtAm” are:
His sun like eye that is full of anger against hiranyakasipu.
His brahmatva or absolute supremacy which frightens the asurAs and enables him to punish them.
His actions, signified by sacrifice, that destroy the asurAs. BhagavAn’s deeds are often referred to as sacrifice, tapas, etc in the shAstra. Because his acts are a means by which he saves his devotees, just as yajna is a means to an end.
vaamapaarshve umaa devii vishhNuH somo.api te trayaH
yaa umaa saa svaya.n vishhNuryo vishhNuH sa hi chandramaaH (text “Rudra-HridayOpanishad”)
Vaidika Interpretation: On his right (auspicious) side, there is Lakshmi who possesses the fame (of quenching his anger), his all-pervasiveness (by which he appeared in the pillar instantly to save prahlAda) and the nectarine auspicious attributes (somA). The fame (of sousIlyam or granting brahmAnandam) is verily vishNu and this vishNu is verily lustrous (as a result of this).
This is his graceful side. “vAma” also means auspicious or beautiful and hence once again is a play on words. The 3 attributes are:
Lakshmi, who suggested PrahlAda pacify Narasimha and thus saved the worlds from his anger.
His all-pervasiveness by which he appears swiftly at the call of his devotees.
His nectarine auspicious attributes. Soma is a name of vishNu as per the sahasranAmA (sOmapO amrtapa sOmaH). “Uma” means yaShas or fame. It refers to his guNam of mingling with others lesser than him or granting brahmAnandam. “ChandrAmso” occurs in the sahasranAmA and refers to his lustre.
yo rudraH sa svayaM brahmaa yo brahmaa sa hutaashanaH
brahmavishhNumayo rudra agniishhomaatka.n jagat.h .(text “Rudra-HridayOpaniShad”)
Vaidika Interpretation: Which Rudra, well-known in the sAstra as Narasimha, is verily the Veda (signified by Brahma), which Supreme Brahman (Brahma), well-known in the sAstra as Narasimha, is verily of the form of fire (ie, jwAla narasimha). This rudra (destroyer of the disease of samsara) possessor (maya) of “VishNuBrahma”, ie, the jivAtma whose essential nature is great (brahma) due to possession of 8 auspicious qualities beginning with apahatapApma and whose attributive knowledge is all-pervasive in mOksha (vishNu). The Universe is verily full of sacrifice of the form of “nama:” salutations to Brahman (agni) and objects of enjoyment such as long age, wealth, etc (sOmA), which are required for upAsaNa.
BrahmI, Brahma, etc are names occurring in the sahasranAma and Parasara Bhattar’s interpretation for these names were used here. The first “Brahma” refers to the Veda. BrahmI means “he who possesses all that goes by the term brahman – prakrti, purusha, ishvara, jivA, etc and this name is used to interpret “brahmavishNu maya” here.
rudramagnimayaM vidyAdviShnuH somAtmakaH smritaH
agnIShomAtmakam chaivA jagatsthAvara jangamam (HarivamSha 2-125-35)
Meaning: Know that Rudra is full of (ie, immersed in) the sacrifice ie, meditation, offerings of “nama:”salutations and remember that vishNu is of the nature of nectar, ie, characterised by auspicious attributes (sOmAtmaka) to be meditated on by upAsakAs like Rudra. The Universe is verily full of sacrifice of the form of “nama:” salutations to Brahman (agni) and objects of enjoyment such as long age, wealth, etc (sOmA), which are required for upAsaNa.
umA shankarayOgO yaH sa yOgO vishNuruchyatE..( text “Rudra-HridayOpanishad”)
Interpretation: By the meditation (yogO) of his own fame, ie, the sousIlya guNa of mingling with others lesser than himself (umA), (the desire) to perform auspicious acts for his devotees and destroy asurAs (shankara), VishNu delights the jivAs (uchyatE) with (such a nature) of being the definite and indestructible means to liberation (sa yOga).
vyakta.n sarvamumaaruupamavyakta.n tu maheshvaram. (text “Rudra-HridayOpanishad”)
Vaidika Interpretation: All the forms (of bhagavAn) that are manifest (vyaktAn) are of the form (ie, manifestations of his) fame, ie, sousIlyam, desire to mingle with others lesser than himself (umA). nArAyaNa who is the Lord of the Worlds (mahEshwara) is called avyakta because in his avatArAs, his true nature is not manifest to everyone.
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Added on 06 Feb 2015:
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Please refer to the "Rudra" debate
I also got the following information from a brilliant, well versed and well learnt Sri Vaishnava.
The narasimha tApanIya Upanishad contains the following enlightening statement:
rudra-japaka-shatam ekam ekena atharvashirah-shikha-adhyapakena tatsamam atharvashirah- shikha-adhyapaka shatam ekam ekena tapaniyopanishad adhyapakena tatsamam tapaniyopanishad-adhyapaka-shatam ekam ekena mantraraja-adhyapakena tatsamam (~Narasimha pUrva tApanIya upanishad).
This says that the mantrarAja pada stOtram is superior to japam of the Rudram, Atharvashika, Atharvasiras and the Narasimha tApanIya itself. This is similar to “srI rAma rAma rAmEti…” in the sahasranAma, which asserts that all the kalyAna guNams described in the sahasranama can be seen in the name of rAma itself. It does not mean the other portions are inferior, but only means that the mantra rAja pada stOtram is the essence of these portions just as “rAma nAmam” is the essence of sahasranAma.
The above part of Narasimha Purva Tapaniya has been accepted by Advaitins also, and a commentary ascribed to Shankara exists.
Maheshvara Tirtha, an ancient advaitic commentator on Sri Ramayana has interpreted "rudrAya vapuShaye" as a pointer to narasimhAvatAra, while explaining the Aditya Hridayam as a Vishnu-stuti (and not a stuti on sUrya as popularly held today).
And advaitins including Shankara, Sureshvara, etc. have interpreted shvetAshvatara upaniShad in a Vaishnava manner only, and that this shvetAshvatara itself contains mantras from Rudram.
