श्रुतिः - विष्णुनात्तमश्नन्ति विष्णुना पीतं पिबन्ति विष्णुना घ्रातं जिघ्रन्ति ।
Whatever is eaten by Vishnu is eaten, Whatever has been drunk by Vishnu is drunk and whatever has been smelled by Vishnu is smelled.
शाबलशाखा - रुद्रेणात्तमश्नन्ति रुद्रेण पीतं पिबन्ति रुद्रेण घ्रातं जिघ्रन्ति ।
Whatever is eaten by Rudra is eaten, Whatever has been drunk by Rudra is drunk and whatever has been smelled by Rudra is smelled.
शांडिल्यः - वासोभूषणमाल्यादि गन्धं तैलं तथौषधम् । सर्वं भगवते दत्वा उपभुंज्यान्निवेदितम् ॥
Garments, ornaments, perfumes, garlands, oils, medicines - offer all these to God first and then use.
कठश्रुतिः - त्रिगुप्सातमश्नीयाद्यदि पाप्मन् शिवानर्पितं भुंक्ष्व तदैनो भुंक्ष्व मलं भुंक्ष्व विषं भुंक्ष्व क्रिमिं भुंक्ष्व अघं भुंक्ष्व अधो गच्छ गच्छ ।
Consume only what has been offered to Shiva, otherwise you are eating sin, excrement, worms and poison.
Naivedyam offered in puja at home
पद्मपुराणम् - स्वस्य देवस्य यद्दत्तं चरुकं तु न संशयः । तत्प्राशयेत्स्वयं प्राज्ञो नैवान्यस्मै प्रदापयेत् ॥
दत्तात्रेयः - ततः स्वयं तु भुंजीत शुद्धं पुत्रादिभिः सह । निवेदितं च देवाय तच्छेषं चात्मशुद्धये ॥
The naivedyam offered during puja at home should be consumed only by the members of the family and not to be shared with outsiders.
About Shiva Nirmalyam
.............. निर्माल्यं नाश्नीयान्न स्पृशेदपि ।
निर्माल्यभेदाः कथ्यन्ते षड्विधास्तेऽपि तद्यथा ॥
Nirmalyam should not be eaten , not even touched.
What is nirmalyam ?
देवद्रव्यं च देवस्वं नैवेद्यं च निवेदितम् ।
चण्डद्रव्यं च निर्माल्यं तेषां लक्षणमुच्यते ॥
Devadravyam, devaswam, naivedyam, niveditam, chandadravyam, nirmalyam - all these six are called nirmalyas.
What is devadravyam ?
वस्त्रभूषणगन्धाद्यं देवद्रव्यमिति स्मृतम् ।
Garments, ornaments and perfumes like sandal - they are called devadravyas and should not be touched by those who don't have the authority and should not be smelled by anyone.
What is devaswam ?
देवस्वं देवसंबन्धि ग्रामक्षेत्रादि गोधनम् ॥
Land, fields and cows etc. belonging to God are called devaswam. They should not be crossed or sold.
What is naivedyam ?
देवार्थं कल्पितान्नाद्यं नैवेद्यं नाम तत् पुनः ।
Rice etc. prepared to be offered to God is called naivedyam.
What are niveditam and Chandadravyam ?
निवेदिताख्यमुत्सृष्टं चण्डद्रव्यं च तद्गतम् ॥
Naivedyam already offered to God is called niveditam. Then it belongs to Chandeshwara and is called Chandadravya.
What is nirmalyam ?
गर्भागाराद्बहिः क्षिप्तं निर्माल्यं तन्न संस्पृशेत् ।
Garalnd, flowers etc. thrown out of the sanctum sanctorum is called nirmalyam. They should not be touched.
षड्विधं चापि निर्माल्यं नोपयुञ्ज्यात् कदाचन ॥
All the above six types of nirmalyam should not be used.
भोजराजः - षद्विधमपि निर्माल्यं न जिघ्रेन्न लङ्घयेत् नाद्यान्न विक्रीणीयात् । क्रव्यादो भवति भुक्त्वा मातङ्गो लङ्घनेऽसिद्धिराघ्राणे वृकः स्पर्शने स्त्रीत्वमथ चण्डालो विक्रये शवरः ।
Bhojaraja says - garments and ornaments meant for Gods should not be worn by others, not even touched. Perfumes and flowers meant for Gods should not be smelled. Land, fields belonging to them should not be crossed or otherwise dealt in. Offerings meant for them should not be consumed or even smelled. After offering to Shiva, where it is meant for Chandshwara it should not be consumed. Flowers etc. after they are offered to God and taken out should not be even touched. All these constitute sin leading to inauspicious results.
व्यासः - देवासुरनरैः स्थाप्ये लिङ्गे चण्डाय शंभुना । दत्तस्तस्मै न सर्वत्र प्रसादः शिवशासनात् ॥
नाश्नीयात्स्थावरे लिङ्गे चंडाय विनिवेदितम् ॥
In the case of Shivalingas installed permanently by devas, asuras and humans Chandeshwara alone has right over naivedyam offered to Shiva. Such naivedyam should not be eaten by others.
Then what should be done with it ?
दत्तात्रेयः- निर्माल्यं नैव संरक्षेत्कूपे सर्वं च निक्षिपेत् ।
It should be disposed in well / pond etc. after offering to Chandeshwara.
More clarity on what kind of naivedyam offered to Shiva can be consumed
आदित्यपुराण्म् - लिङ्गे स्वायम्भुवे बाणे रत्नजे रौप्यनिर्मिते । सिद्धप्रतिष्ठिते लिङ्गे चण्डभागो न विद्यते ॥
There is no share for Chandeshwara in Swayanbhu lingas, Bana lingas ( found in Narmada ) and those made of gems such as emerald or silver. Even in the case of lingas consecrated by Siddhas there is no share for Chandeshwara. Naivedyam offered to such lingas can be consumed. This should cover lingas at home.
शिवनिर्माल्यवत्सौरं चण्डे दत्तं ततोऽधिकम् ।
Nirmalyam of Surya should be also treated like Shiva nirmalyam.
गुरु - पुस्तक - वह्नि - यक्ष - नाग - योगीन्द्र - गण - मातृ - गौरीषु शिवनिर्माल्यवत् न भवति ।
These rules do not apply to nirmalyam of Guru, Granthas, Agni, Yaksha such as Kubera, Yogindra, Ganesha and Devis.
To sum it up -
1. Prasada from public Shiva temples except when they are Swayambhu, bana linga, made of gems or silver or installed by Siddhas or of the movable type should not be eaten. The used flowers should not be touched.
2. There is no such restriction for Shiva puja at home.
3. There are no such restrictions for other deities.
4. Naivedyam offered in puja at home should not be shared with others.
5. One should not misuse devadravyam or devaswam.