Since the total space of a cover is locally homeomorphic to the base space, local topological properties (like local (path) connectedness, T1 etc.) lift from the base space to the total space. The same holds for local compactness, if we assume the base space is Hausdorff.
My question is, given a non-Hausdorff locally compact space $X$, must every cover of $X$ be locally compact.
My intuition says no, because the compact neighbourhoods in $X$ might be too big to be "seen" by the covering structure. I can't think of any counterexamples, mainly because I don't remember any non-compact non-Hausdorff spaces right now and won't have access to Steen & Seebach for a couple of days. Although, when writing this, it strikes me that there could be conditions on $X$, which would ensure that the covering projection is a proper map, in which case we would be done.
NB: By a locally compact space I mean a space in which every point has a compact neighbourhood.
Edit: Thinking further, if the base space is T3/regular (the weaker of the two, whatever your convention might me) and locally compact every cover is locally compact, basically for the same reason as in the Hausdorff case.
Best Answer
As mentioned in the comments, this question was subsequently asked on MathOverflow. It was answered there by Taras Banakh (see his answer there), so I am writing this answer to close this version as well.