Why people are not allowed to take images of god/goddesses in Garbha gudi (sanctum sanctorum) of a temple. Is it a sin according to scriptures? or any other reason behind such restriction?
In addition to the above, I also heard that some mantras should not be published anywhere and should not be printed on papers, and should not be digitized. why?
Photography of the murthis are banned to protect them from ridicule. Christian missionaries and western audiovisual media have taken pictures of the images and ridiculed them. Christian missionaries in the 19th century used to beat murthis with stick in public to prove to Hindus that these murthis are not gods. This is one reason for the current restriction.
Some people do not want to publish the texts of some mantras because they are afraid that such mantras may be used for nefarious purposes by unscrupulous people.
Mantras are to be recited with perfect grammar else it may mean something else. The vibrations which are produced by chanting the vedic mantras are very powerful and said to heal many inner & outer diseases. So one has to learn them from masters who know the complete meaning & purpose the the mantra. They have to be learnt from person to person with correct pronunciation.
Temples are the spiritual centers. One learns how to build character through the puranas or the speeches delivered there. One learns the joy of sharing by doing charity or learns from others who does there. Soft skills which are presented to please god like singing (music heals), dancing (reasoning skills), poetry (imaginative & creative skills) will take us to next level both spiritually and also with worldly knowledge. Temples energize us like this in numerous ways which may be difficult to explain all in one answer.
Idols of god are installed with vedic hymns in Garbha gudi, the process is called prana pratista meaning after that process god Himself/Herself lives in that murthi. Everyday god takes different upacharas from suprabatham to unjal seva through that murthi. This creates the divine environment around the temple and one who visits temple gets peace and confidence in account of unrecognized energy set around. So, if such powerful god is taken photo, kept in home, people may become lazy to visit temple often they may convince themselves on seeing the image taken.
Mantra sadhana or the practice of using mantras, is a practice that depends completely on phonetics and is primarily meant to be transmitted in oral form. Writing down a mantra suffers from the issues of transliteration mistakes, incorrect pronunciation by the final recipient and so forth. That said there is no "religious" significance of preventing photography per se. Different temples and shrines set up different rules depending on their beliefs and opinions on appropriate conduct with respect to their particular deity. For example, some sects might be desirous of preventing people from displaying pictures taken in settings not in line with the deity's followers' beliefs.
About the Mantra: With my discussion on "Why Gayatri Mantra restricted by Vishwamitra, so that other can not use?" it is said that there lots of thing in this mantras than just peace and good health. Mantra can also be used as Weapon, Mantra can be used for bad thing like for MARAN vidya. So, many of the mantras should must be kept in secret so that can not used for harmful purpose.
If we talk about Vedas mantra than, these mantra should be pronounce at appropriate time and with appropriate RAGA otherwise, you can not get vibration of mantra and may be you lost faith in Mantra.
And in last, on my discussion of above question of Gayrti Mantra, one sadhu(bava) told me that there is a reverse spell in Gaytri Mantra, which is related with BRAMASHTRA weapon.
