I have heard that the yajnopavitam cannot be removed ever after once initiated. However, I have also heard Ramakrishna Paramahamsa removed his own one countless times. What do the scriptures say about the removal of the yajnopavitam after initiation?
As for the first question (or the question which the title asks) the answer is NO (apparently).
A twice-born one should always wear the sacred thread and tie up the tuft of hair on his crown. He should put on a most excellent white piece of cotton or silk cloth without any hole, but quite different from the one used before
Usana Smriti's chapter 1 verses.
He should, daily, study the Vedas and put on his sacred thread. He should speak the truth and control his passion. [Such conduct] leads to the attainment of emancipation.
Yet another one:
One must always put on the sacred thread and bind the tuft of hair on the crown : what one does without having the sacred thread on or without binding the tuft of hair on the crown, is equal to an act not done.
Katyayana Smriti 4.1
However, there are some verses, like the one given immediately below, which suggest, wearing the sacred thread, at all the times, was not a prescribed rule.
Being, commanded with " Kurushwa" (do it), he should, with the sacred thread on,* offer oblations [tthe Fire.] Wearing the sacrificial thread and with the Kusa in hand, one should perform the Homa. (Usana Smriti)
The commentary on this verse is as follows:
This and similar other passages distinctly show that to put on the sacred thread always, was not the practice prevailing then. But the Brahmanas used to put on the sacred thread, as a distinctive mark* before the performance of any particular religious rite.
Anyways, coming to Sri Ramakrishna now. You are right about the fact that he used to remove it at times, during his meditative sessions to be precise.
The book " Short life of Sri Ramakrishna by Swami Tejasananda" , which i have a copy of (in Bengali though) mentions the story which goes like this:
During the nights, after the temple used to get closed, Sri RAmakrishna used to go to the famous Panchabati forest and used to meditate upon Mother Goddess under a tree there. This was his regular habit. One day, as usual, after Sri RAmakrishna has gone for meditation, his nephew Hriday, followed him and silently entered that place. He was shocked by what he saw there; He saw his mAmA or Thakur had removed not only the sacred thread but also his clothes and was deeply engrossed in DhyAna in that naked condition. On seeing that Hriday thought to himself " Mama must have gone mad! " and started calling him. After some effort Sri RAmakrishna regained consciousness.
When Hriday enquired about what he was doing there naked, Sri RAmakrishna replied "Do you know, when one meditates one needs to meditate in this manner-without any bondage (Tui ki janis? Eirup pasamukto hoi dhyan korte hoi). From birth a man is bound by the eight pAshas viz - Ghrina, LajjA, Kula, shila, Bhaya, MAna, JAti and AbhimAna. Even this Yajnopavita is a pAsha because it reminds me always that "I am Brahmin and hence i am superior. This pride of birth is also one such hindrances. He further said " When one mediates on the Mother one should be from all kinds of bondage and do it with single-minded concentration, otherwise one can not attain Her (MAke dakte hole ei sob pas phele diye ek mone dAkte hoi, tai esob khule rekhechi, DhyAn korA shesh hole phirbar somi AbAr porbo".
From Chapter 5 of the aforementioned book.
Now, these 8 PAshas are also widely known in the Agamas. And, the soul which is still bind with 8 of them is called the Pashu. For example, the following are the 8 pAshAs as mentioned by Lord Shiva:
GhrinA shankA bhayam lajjA jugupsA cheti panchami | Kulam shilam tathA jAtir ashto pAshAh prakirtitA ||
Disgust, apprehension, fear, shame, feeling of insult, Kula-Shila-JAti (pride of family etc) are the eight bondages (pAshAs).
KulArnava Tantram 13.90
In realizing the ultimate goal these 8 PAshas are like major hindrances for a spiritual aspirant. And the Yajnopavita will fall under the Kula-JAti PAshA category whereas the clothes will come under the Shame-PAshA. So, that's what the purport of Sri RAmakrishna's message to his nephew is.
Even the Smritis don't deny the fact that a Brahmin who is devoid of the knowledge of Self but takes pride in his sacred thread is nothing but a Pashu.
A Vipra who does not know the true nature and being of Brahman but is always proud of his sacrificial thread, is for that sin called a Pasu (beast).
So, do not judge an extraordinary Yogi-Saint of the order of Sri RAmakrishna from the rule books. He possibly can avoid at ease what are some must-follow rules for any other ordinary Brahmins.