I read guitar tabs recently and was confused by the "let ring" concept. If I'm not using palm mute, shouldn't it be ringed instead? E.g, I change C chord to D chord, 5th string still ringing and shouldn't be played in D chord. Should I mute it before playing D chord?
It would help to have some better examples. In the example you gave the 5th string open is A and that's okay to put on a D chord. However, I can't imaging how that is ringing since the C chord has the 3rd fret of the 5th string fingered.
I am used to seeing "let ring" when they want you to play something arpeggiated. For example the into to Stairway to Heaven has the guitar playing a series of chords one note at a time, like Amin = (x, x, 7, 5, 5, 5) where x means don't play and the other numbers are fret numbers on the strings 6 through 1. In this case "let ring" means let each note ring as the others are being played so that the entire chord fills out.
In your case if you are just playing chords one after another it does not make sense to let notes from the previous chord continue as that could conflict.
If the open A is ringing because you released the finger and that naturally acts as an attack then you have a different issue. In this case I'd say the instructions are trying to tell you to let each chord ring, but you still need to play each with clean technique. So when one chord is over and the next one starts you do need to control the extra open sting sounds. This may require you to mute with either hand, at least momentarily.
Lastly, the open A has harmonics that align with those of the D chord. So it is possible that even with clean technique on an acoustic guitar when you play the D chord the A string starts vibrating. This is completely nature and, unless you don't like it, should be left alone. The sympathetic resonances between the open strings help fill the chords out and make the overall sound richer.
Instruments that have mutiple elements for producing notes, such that those elements keep producing tone after they are activated:sustain—plucked strings, struck blocks and such, support a technique called lasciare suonare ("let ring"). This means that we play the music by initating all the right notes at the right time, but allow them to decay naturally rather than dampening them.
On a piano, music can be played lasciare suonare by holding down the sostenuto pedal.
Under this technique, it doesn't matter that some of the notes don't go together in terms of harmony; the dissonances you hear contribute to the music. The beauty of lasciare suonare is that it makes simple passages sound rich and complicated. It can create a "haunting" feeling.
Every wind-up, plucked-reed music box plays lasciare suonare, which is responsible for the charming sound, and a feeling of reverberation that adds depth to the tiny sound source.
The magic of the harp is also due to lasciare suonare.
Most vibraphone/metallophone/xylophone music is lasciare suonare, too.
Note that even when you do not play lasciare suonare, if you merely play legato, or even just close to the full time value of the individual notes, successive notes that are produced by different strings will still bleed into each other, due to the decay caused by damping not being instantaneous.
On a piano, when the sostenuto pedal is not being used, all notes have a release time: the time required for the second decay that begins when the player lifts the finger from the key, to the time when the note stops. This means that if you play a quick passage of notes on a piano, they bleed into each other; and the faster you play them, the more it is so. The felt damper which stops the string cannot do its job instantaneously, and it doesn't get any faster with increased tempo. Moreover, it would be odd if it could do so.
People who program synthesizers understand that a note's amplitude envelope has four stages: attack, sustain, decay, and release. Decay is the gradual deadening of the note after the initial attack while the note is being sustained. Release is the second decay (usually much faster), when the note is stopped.
On a decently programmable synthesizer, it is possible to tweak the piano patch such that the notes have a release time of zero (the sound stops instantly when the key is released without any additional decay). I tell you, it becomes weirdly choppy sounding, almost not like a piano. All imperfections in technique seem laid bare.
In practice, let your ears decide! In theory, is the note part of the chord you are trying to carve? If not, mute. You'll know when to mute with experience. There are techniques that use either hand for muting, both for finger style and pick.
In your specific example the 5th string is A, which is the 5th of D, part of the major triad, so not muting it sound perfectly fine. The 6th is an E, not part of the triad, so you might want to be a little more careful with it. It's common to do the second fret, which is the major third, part of the triad again. You can strum all string that way. You can also strum all strings by muting the 6th with your left hand's thumb, if the neck is small enough.
But it's mostly your ears and experience. Play with all possibilities, and practice in a way that you can do either muted and not muted whenever you feel like. That E in the 6th string can sound great in the right contexts, same that you'd expect with any note in any harmony. Diatonic second, so not even that big of a leap!