First, let me cite skepticism that $L \neq NL$. As it has been shown that undirected graph connectivity is in $L$ (Reingold), and that $NL=coNL$ (Immerman-Szelepcsényi), I think that confidence in $L \neq NL$ has only decreased. Some prominent researchers have never had a strong belief. For example, Juris Hartmanis (founder of the CS department at Cornell and Turing award winner) has said:
We believe that NLOGSPACE differs from LOGSPACE, but not with the same depth of conviction as for the other complexity classes. (Source)
I know he said similar things in the literature as far back as the 70's.
There is some evidence against $L=NL$, although it is circumstantial. There has been work on proving space lower bounds for $s$-$t$ connectivity (the canonical $NL$-complete problem) in restricted computational models. These models are strong enough to run the algorithm of Savitch's theorem (which gives an $O(\log^2 n)$ space algorithm) but are provably not strong enough to do asymptotically better. See the paper "Tight Lower Bounds for st-Connectivity on the NNJAG Model". These NNJAG lower bounds show that, if it's possible to beat Savitch's theorem and even get $NL \subseteq SPACE[o(\log^2 n)]$, one will certainly have to come up with an algorithm that's very different from Savitch.
Still, I don't know of any unlikely, unexpected formal consequences that come from $L=NL$ (except for the obvious ones). Again, this is primarily because we already know things like $NL=coNL$.