The three existing answers here are all good and correct answers and all say more or less the same thing: it's still technically a C chord but it might sound weird.
So in general you shouldn't try to sound weird.
But there are also times when you COULD use that voicing and times when you SHOULD use that voicing.
You COULD use that voicing if you're not responsible for playing the bass note ie you have a bass player. Then your lowest note is just another note on top of the bassist's bass note and your low E might actually contribute to a fuller sounding harmony. This is not the interesting part though.
The interesting part is that there are also times when you SHOULD use that particular voicing. These will be in the situation where you ARE responsible for providing the bass note.
Aaron mentions that "the decision on whether or not to include the low E will depend more on the musical context and whether or not that particular sound "fits" with whatever is happening before and after."
Tim gives a specific example: "The open bottom E version works as a transitionary voicing just before an F chord - that E is the leading note, so moves up to 1st fret bottom string when F gets played, and sounds good."
What they're both describing here is a situation where the E is part of a melodic figure in the bass. In Tim's example of a C in first inversion going to an F chord in root position, the bass melody is E-F. While there is no need for the melody to be stepwise, another very common use of the voicing would be on a guitar in drop D tuning going F in root position to C in first inversion (with E in the bass) to a D minor chord in root position. That progression has the stepwise melody F-E-D in the bass. The same progression in very common in other keys in standard guitar tuning (eg G D/F# Em)
Similarly another C major voicing mentioned by Tim (332010) is useful in other situations. With this voicing you can pluck a root-fifth-root-fifth bass line while strumming the higher notes for harmony. You can now start to imagine how to use various voicings of chords to construct flowing bass melodies.
Obviously myriad other melodies are possible with the bass line using all the available chord voicings. The point is that the use of a voicing is absolutely appropriate if used deliberately and precisely.
In the end, when it comes to music trust your ear over your brain.