That's right, you can definitely play the open A string when playing a D chord. It's not necessary to mute it. A is part of the D major chord (D F# A), so it will technically be a Second Inversion. (The F# in the bass makes it First Inversion.)

It's also true that the F# and open A will sound slightly muddy together (because they're a minor 3rd apart in a low register), but that's a matter of taste and it depends on the situation.

Sometimes it's best to play fewer notes in a chord when in a band situation, since so many other instruments are filling in frequencies.