Run the SQL procedure unanswered

The question is not very clear, so I'm going to try to rephrase - if I'm close, great; if not, let me know. I think what you're asking is essentially to run a command (via a stored procedure), and have the code continue regardless of whether the command completes successfully or not. If that is the case, then simply use a try/catch region to swallow the exception. Neither dapper nor ADO.NET expose something more specific.

try {
    conn.Execute("MyProc",
        new { id: 42, name: "abc" }, // parameters to the proc
        commandType: CommandType.StoredProcedure);
} catch { /* swallow it down, oh so tasty! */ }