This largely is based on the context that you are playing the chords in. If you are just playing for yourself, then you can definitely use whatever chord voicings you like. If you are trying to play someone else's songs, like learning a cover song, if you play those extra notes, you're not really playing what the songs is. This can be ok if you're not trying to play the song exactly like the original. In some cases, you and/or the audience may never notice the difference between the original and the change in voicing that you've decided upon.
If you are trying to emulate a particular style of music, changing voicings may pose a larger problem, particularly for voice leading. If you're not familiar with the rules of voice leading, the general idea is that there are "rules" for how to go about voicing chords and how those voices move from chord to chord, which allow the different voices of the chords to remain independent. Proper voice leading can lead to a richer sounding harmony and by breaking these conventions, it can take away from the fullness/richness of the harmony in general. This is most common in Classical music but you can certainly find other styles of music that employee these voice leading rules. You'll find most often that proper voice leading on guitar is best accommodated by the open chord voicings. One or two notes that are in the chord may be dropped to maintain proper voice leading, so adding these back into the mix can subvert voice independence and remove the richness of the harmony. This is something that a newer player or someone that mostly plays rock and pop would likely not notice but can make a fairly big difference in the overall feel of the harmony.
You also want to pay attention to your bass note, as Peter mentioned in his answer. Changing the lowest note of your chord, called an inversion, can drastically change the feel of a given chord, particularly if you are putting the third in the bass, which feels much less stable. A good example of this would be the open C voicing, where you typically mute the low E string, even though the E is part of the chord. If you add the E in, the chord will usually feel less stable and more dissonant. There are also what we call Functions for different chords and inverting chords can affect the Function or the effectiveness of the Function. Back to the C/E example, this chord would have a stronger desire to resolve to F than a normal C chord, so it would work best in a situation where it moves to F. There are plenty of ways to incorporate inversions but it's a good idea to be aware of how it is affecting the overall feel of the harmony.
You should also be mindful of whether or not you're playing with a bass player, or another instrument that may be playing a lower note than your lowest. If the bass player is playing a low C, then the overall chord would be describe just as C, not C/E but some guitar players may still refer to the voicing of the chord as C/E. If the bass player is playing the same octave as the C you are playing, and then you decide to add the E in the bass, you're effectively changing the harmony to C/E because the bass player is actually not playing the lowest note anymore. You can also run into chord voicings getting a little muddy sounding if the note you're adding into the mix is particularly close to what the bass player is playing, so you'll want to pay attention to that.