For discrete wavelet transforms, is redundancy needed to ensure perfect reconstruction?
Theoretically, the DWT (or the discrete wavelet transform) is (under some conditions) a discrete sampling of the continuous  wavelet transform allowing perfect synthesis, for "infinite length" discrete data. But this yields "discrete" infinitely many coefficients.
In some cases, There exist some wavelets, like Haar's, that can be implemented in a non expansive manner.

Practically, for discrete length signals that are not "even" (like with a power-of-two length), one must find options related to signal and wavelet symmetries. This is an important issue in data compression, as one often does not want to increase data size before entropy coding (but this is another debate). 

To remain concrete, think about signal $[1,-1,3]$. How would you represent it with only 3 coefficients, knowing that the dyadic version should have the same number of approximation and detail coefficients? Many tricks have been developed for that. However, if you are not concerned with compression, most can afford a little redundancy.