To support emojis when using MySQL in .NET Core (version 2.2 in my case) you need to do a few things.
First of all, add CharSet=utf8mb4; to your database connection string. In other words, your code in Startup.cs should look something like this.
string db = @"Server=domain.com;Database=db;Uid=user;Pwd=pass;CharSet=utf8mb4;";
services.AddDbContext<AppDbContext>(options => options.UseMySQL(db));
You also need to make a couple of database changes. The charset needs to be set to utf8mb4 both for the column you are storing the emojis in and as the default for the entire database.