Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a logconcave function on $U$ (i.e., bounded and realvalued). Under what conditions does $f$ have a continuous extension to the closure $\overline U$?

I do not see how $\log$concavity should imply any form of continuity. For instance, if $\\cdot\$ is any seminorm on the locally convex space $X$ then $f(x) = e^{\x\}$ will be bounded and $\log$concave but it will only be continuous if the seminorm is continuous. What you really want in order to be able to extend $f$ to the boundary of $U$ is uniform continuity and for that the fact that $f$ is $\log$concave may help but is certainly not sufficient. 

