Are there any open source SAT solvers with UNSAT core extraction algorithm built in?
MUSer2 is probably the tool for extracting minimal unsatisfiable cores (MUS) currently. It treats the SAT solver as a black box, so you can plug in any solver you want.

If you want a solver with MUS extraction capability, the newest version of PicoSAT comes with a utility tool PicoMUS that does the job.