On sparse complete sets and P vs L Mahaney's Theorem tells us that if there is a sparse $NP$-complete set under polynomial-time many-one reductions, then $P = NP$.  (See "Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis")

Are there known consequences of the existence of sparse complete sets for other complexity classes?  In particular, if there is a sparse $P$-complete set under logspace many-one reductions, does that imply $P = L$?
Yes, exactly what you suggested is true: if there is a sparse $\mathbf{P}$-complete set under log-space many-one reductions, then $\mathbf{P} = \mathbf{L}$. This was conjectured by Hartmanis in 1978 and proven by Cai and Sivakumar in 1995. See this paper.

Hartmanis also conjectured that if there is a sparse $\mathbf{NL}$-complete set under log-space many-one reductions, then $\mathbf{NL} = \mathbf{L}$. This was also proven by Cai and Sivakumar in 1997; see this other paper.