Consequences of sub-exponential proofs/algorithms for SAT
If SAT had a subexponential-time algorithm, the you would disprove the exponential time hypothesis. 

For fun consequences: if you showed that circuit SAT over AND,OR,NOT with $n$ variables and $poly(n)$ circuit gates can be solved faster than the trivial $2^n poly(n)$ approach, then by Ryan Williams' paper you show that $NEXP \not\subseteq P/poly$.