Some SAT solvers and SMT solvers offer an interface that lets you push clauses, and then later pop/retract them and push some new ones. You could explore to see whether this offers a speedup in your situation. There are no guarantees, and the only way to tell is to try it.
There are no competitions targeting general integer programming or mixed integer programming, but there are (or were) benchmarks, such as the MIPLIB (linear) and the MINLPLIB (nonlinear).
There are competitions for subsets (PB, SAT, max-SAT) and for constraint programming, as you and other answers pointed out. You can find many competitions (DIMACS ...
There are competitions for constraint satisfaction solvers. Some problems there can be readily translated to IP solvers as well. See e.g., MiniZinc challenge which has taken place yearly since 2008 or the XCSP competition.
There was a Pseudo-Boolean solver competition from 2005-2012, but (as far as I can tell) nothing since then. Integer Linear Programming is a subset of Pseudo-Boolean programming. See the 2012 competition page for results and links to other competition results.