What type of database server/service to use for an increased insert rate? Currently I am using MySQL and it processes 10,000 inserts per day, and soon it is going to be 200,000 inserts per day.

Now I am using a normal server which is okay with 10,000 inserts but processing 200,000 will require a large DB server. So my question is, what type of database server will I need to migrate to that could manage such a large insert rate?
What makes you think 200,000 inserts a day will be a problem? That is about 140/minute so if they are reasonably sized and spread evenly through the day just about any database will cope on the most modest hardware. If most of them are crammed into one hour than that would be approx 55 per second, so knowing how the pattern of your apps activity is likely to vary is vitally important when trying to predict how it will scale. The size of the inserts makes a difference too: a few hundred bytes per row, or LOB fields with several Mb of data each?

As it stands this question will get closed as "too broad" as it is essentially a "how long is a piece of string" one.

The only thing we can really suggest with this little information is "test it and see what happens".