No, they don't. We basically design them so that they cannot preserve distances. An autoencoder is a neural network which learns a "meaningful" representation of the input, preserving its "semantic" features. The quoted words (like so many terms in Deep Learning papers) have no rigorous definition, but let's say that, trained on a set of inputs, the autoencoder should learn some common features of these inputs, which allow to reconstruct an unseen input with small error 1.
The simplest way for the autoencoder to minimize the differences between input and output (reconstructed input) would be to just output the input, i.e., to learn the identity function, which is an isometry, thus it preserves distances. However, we don't want the autoencoder to simply learn the identity map, because otherwise we don't learn "meaningful" representation, or, to say it better, we don't learn to "compress" the input by learning its basic semantic features and "throwing away" the minute details (the noise, in the case of denoising autoencoder).
To prevent the autoencoder from learning the identity transformation, and forcing it to compress the input, we reduce the number of units in the hidden layers of the autoencoder (bottleneck layer or layers). In other words, we force it to learn a form of nonlinear dimensionality reduction: not for nothing, there is a deep connection between linear autoencoders and PCA, a well-known statistical procedure for linear dimensionality reduction.
However, this comes to a cost: by forcing the autoencoder to perform some kind of nonlinear dimensionality reduction, we prevent it from preserving distances. As a matter of fact, you can prove that there exists no isometry, i.e., no distance preserving transformation, between two Euclidean spaces $\mathbb{E}^n$ and $\mathbb{E}^m$ if $m < n$ (this is implicitly proven in this proof of another statement). In other words, a dimension-reducing transformation cannot be an isometry. This is quite intuitive, actually: if the autoencoder must learn to map elements of a high-dimensional vector space $V$, to elements of a lower-dimensional manifold $M$ embedded in $V$, it will have to "sacrifice" some directions in $V$, which means that two vectors differing only along these directions will be mapped to the same element of $M$. Thus, their distance, initially nonzero, is not preserved (it becomes 0).
NOTE: it can be possible to learn a mapping of a finite set of elements of $V$ $S=\{v_1,\dots,v_n\}$, to a finite set of elements $O=\{w_1,\dots,w_n\}\in M$, such that the pairwise distances are conserved. This is what multidimensional scaling attempts to do. However, it's impossible to map all the elements of $V$ to elements of a lower-dimensional space $W$ while preserving distances.
1things gets more complicated when we refer to my favourite flavour of autoencoder, the Variational Autoencoder, but I won't focus on them here.